This guest blog entry concerns the many roles a World Digital Mathematical Library (WDML) could play for the mathematical community worldwide. We seek input to help sketch how a WDML could be so much more than just a huge collection of digitally available mathematical documents. If this is of interest to you, please read on!

The “we” seeking input are the Committee on Electronic Information and Communication (CEIC) of the International Mathematical Union (IMU), and a special committee of the US National Research Council (NRC), charged by the Sloan Foundation to look into this matter. In the US, mathematicians may know the Sloan Foundation best for the prestigious early-career fellowships it awards annually, but the foundation plays a prominent role in other disciplines as well. For instance, the Sloan Digital Sky Survey (SDSS) has had a profound impact on astronomy, serving researchers in many more ways than even its ambitious original setup foresaw. The report being commissioned by the Sloan Foundation from the NRC study group could possibly be the basis for an equally ambitious program funded by the Sloan Foundation for a WDML with the potential to change the practice of mathematical research as profoundly as the SDSS did in astronomy. But to get there, we must formulate a vision that, like the original SDSS proposal, imagines at least some of those impacts. The members of the NRC committee are extremely knowledgeable, and have been picked judiciously so as to span collectively a wide range of expertise and connections. As president of the IMU, I was asked to co-chair this committee, together with Clifford Lynch, of the Coalition for Networked Information; Peter Olver, chair of the IMU’s CEIC, is also a member of the committee. But each of us is at least a quarter century older than the originators of MathOverflow or the ArXiv when they started. We need you, internet-savvy, imaginative, social-networking, young mathematicians to help us formulate the vision that may inspire the creation of a truly revolutionary WDML!

Some history first.  Several years ago, an international initiative was started to create a World Digital Mathematical Library. The website for this library, hosted by the IMU, is now mostly a “ghost” website — nothing has been posted there for the last seven years. [It does provide useful links, however, to many sites that continue to be updated, such as the European Mathematical Information Service, which in turn links to many interesting journals, books and other websites featuring electronically available mathematical publications. So it is still worth exploring ...] Many of the efforts towards building (parts of) the WDML as originally envisaged have had to grapple with business interests, copyright agreements, search obstructions, metadata secrecy, … and many an enterprising, idealistic effort has been slowly ground down by this. We are still dealing with these frustrations — as witnessed by, e.g., the CostofKnowledge initiative. They are real, important issues, and will need to be addressed.

The charge of the NRC committee, however, is to NOT focus on issues of copyright or open-access or who bears the cost of publishing, but instead on what could/can be done with documents that are (or once they are) freely electronically accessible, apart from simply finding and downloading them. Earlier this year, I posted a question about one possible use on MathOverflow and then on MathForge, about the possibility to “enrich” a paper by annotations from readers, which other readers could wish to consult (or not). These posts elicited some very useful comments. But this was but one way in which a WDML could be more than just an opportunity to find and download papers. Surely there are many more, that you, bloggers and blog-readers, can imagine, suggest, sketch. This is an opportunity: can we — no, YOU! — formulate an ambitious setup that would capture the imagination of sufficiently many of us, that would be workable and that would really make a difference?

The classical Brauer-Fowler theorem asserts that if a group has many involutions, then it must have a large non-trivial subgroup:

Theorem 1 (Brauer-Fowler theorem) Let be a finite group with at least involutions for some . Then contains a proper subgroup of index at most .

This theorem (which is Theorem 2F in the original paper of Brauer and Fowler, who in fact manage to sharpen slightly to ) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if is a an involution of a group , and the centraliser has order , then clearly (as contains and ) and the conjugacy class has order (since the map has preimages that are cosets of ). Every conjugate of an involution is again an involution, so by the Brauer-Fowler theorem contains a subgroup of order at least . In particular, we can conclude that every group of even order contains a proper subgroup of order at least .

Another corollary is that the size of a simple group of even order can be controlled by the size of a centraliser of one of its involutions:

Corollary 2 (Brauer-Fowler theorem) Let be a finite simple group with an involution , and suppose that has order . Then has order at most .

Indeed, by the previous discussion has a proper subgroup of index less than , which then gives a non-trivial permutation action of on the coset space . The kernel of this action is a proper normal subgroup of and is thus trivial, so the action is faithful, and the claim follows.

If one assumes the Feit-Thompson theorem that all groups of odd order are solvable, then Corollary 2 suggests a strategy (first proposed by Brauer himself in 1954) to prove the classification of finite simple groups (CFSG) by induction on the order of the group. Namely, assume for contradiction that the CFSG failed, so that there is a counterexample of minimal order to the classification. This is a non-abelian finite simple group; by the Feit-Thompson theorem, it has even order and thus has at least one involution . Take such an involution and consider its centraliser ; this is a proper subgroup of of some order . As is a minimal counterexample to the classification, one can in principle describe in terms of the CFSG by factoring the group into simple components (via a composition series) and applying the CFSG to each such component. Now, the “only” thing left to do is to verify, for each isomorphism class of , that all the possible simple groups that could have this type of group as a centraliser of an involution obey the CFSG; Corollary 2 tells us that for each such isomorphism class for , there are only finitely many that could generate this class for one of its centralisers, so this task should be doable in principle for any given isomorphism class for . That’s all one needs to do to prove the classification of finite simple groups!

Needless to say, this program turns out to be far more difficult than the above summary suggests, and the actual proof of the CFSG does not quite proceed along these lines. However, a significant portion of the argument is based on a generalisation of this strategy, in which the concept of a centraliser of an involution is replaced by the more general notion of a normaliser of a -group, and one studies not just a single normaliser but rather the entire family of such normalisers and how they interact with each other (and in particular, which normalisers of -groups commute with each other), motivated in part by the theory of Tits buildings for Lie groups which dictates a very specific type of interaction structure between these -groups in the key case when is a (sufficiently high rank) finite simple group of Lie type over a field of characteristic . See the text of Aschbacher, Lyons, Smith, and Solomon for a more detailed description of this strategy.

The Brauer-Fowler theorem can be proven by a nice application of character theory, of the type discussed in this recent blog post, ultimately based on analysing the alternating tensor power of representations; I reproduce a version of this argument (taken from this text of Isaacs) below the fold. (The original argument of Brauer and Fowler is more combinatorial in nature.) However, I wanted to record a variant of the argument that relies not on the fine properties of characters, but on the cruder theory of quasirandomness for groups, the modern study of which was initiated by Gowers, and is discussed for instance in this previous post. It gives the following slightly weaker version of Corollary 2:

Corollary 3 (Weak Brauer-Fowler theorem) Let be a finite simple group with an involution , and suppose that has order . Then can be identified with a subgroup of the unitary group .

One can get an upper bound on from this corollary using Jordan’s theorem, but the resulting bound is a bit weaker than that in Corollary 2 (and the best bounds on Jordan’s theorem require the CFSG!).

Proof: Let be the set of all involutions in , then as discussed above . We may assume that has no non-trivial unitary representation of dimension less than (since such representations are automatically faithful by the simplicity of ); thus, in the language of quasirandomness, is -quasirandom, and is also non-abelian. We have the basic convolution estimate

(see Exercise 10 from this previous blog post). In particular,

and so there are at least pairs such that , i.e. involutions whose product is also an involution. But any such involutions necessarily commute, since

Thus there are at least pairs of non-identity elements that commute, so by the pigeonhole principle there is a non-identity whose centraliser has order at least . This centraliser cannot be all of since this would make central which contradicts the non-abelian simple nature of . But then the quasiregular representation of on has dimension at most , contradicting the quasirandomness.

— 1. Character-based proof —

Now we give the character-based proof of Theorem 1, following Isaacs. We assume familiarity with the basic theory of characters, as reviewed in this recent blog post.

Let be a finite group, and let be a character of associated to some finite-dimensional unitary representation , thus . We can then consider the tensor square representation defined in the usual manner:

One easily checks that this representation has character

On the other hand, the tensor square representation splits into the symmetric part and the alternating part , since the symmetric and alternating portions of the tensor square are preserved by the action of . Thus we have a splitting

in particular, taking inner products with the trivial character (i.e. computing the dimension of the invariant component of all representations listed above) we conclude that

noting that the right-hand side vanishes if is not real (so that is orthogonal to ). On the other hand, we can compute the character of explicitly using an orthonormal basis for , which induces an orthonormal basis , for . Then the character is equal to

which after some algebra (using symmetry to eliminate the constraint and noting that ) simplifies to

Inserting this back into (1) we obtain the bound

In particular, if is irreducible, its norm is and we conclude the following bound of Frobenius and Schur:

(Indeed, this argument shows that the expression is either , , or and vanishes unless is real, although we will not need these additional facts here.) Now from the orthogonality of irreducible characters we have

so if we average this in and use (2) we conclude that

since for at least values of (indeed we have values if we also add the identity). On the other hand, from (3) we have the well known identity

so from Cauchy-Schwarz (after subtracting off the trivial character) we have

But the right-hand side is the number of non-trivial conjugacy classes of , so by the pigeonhole principle there is a non-trivial conjugacy class with cardinality at most , which gives a centraliser of order at least , as required.

for all ; by anti-symmetry one can also rewrite the Jacobi identity as

We will usually omit the subscript from the Lie bracket when this will not cause ambiguity. A homomorphism between two Lie algebras is a linear map that respects the Lie bracket, thus for all . As with many other classes of mathematical objects, the class of Lie algebras together with their homomorphisms then form a category. One can of course also consider Lie algebras in infinite dimension or over other fields, but we will restrict attention throughout these notes to the finite-dimensional complex case. The trivial, zero-dimensional Lie algebra is denoted ; Lie algebras of positive dimension will be called non-trivial.

Lie algebras come up in many contexts in mathematics, in particular arising as the tangent space of complex Lie groups. It is thus very profitable to think of Lie algebras as being the infinitesimal component of a Lie group, and in particular almost all of the notation and concepts that are applicable to Lie groups (e.g. nilpotence, solvability, extensions, etc.) have infinitesimal counterparts in the category of Lie algebras (often with exactly the same terminology). See this previous blog post for more discussion about the connection between Lie algebras and Lie groups (that post was focused over the reals instead of the complexes, but much of the discussion carries over to the complex case).

A particular example of a Lie algebra is the general linear Lie algebra of linear transformations on a finite-dimensional complex vector space (or vector space for short) , with the commutator Lie bracket ; one easily verifies that this is indeed an abstract Lie algebra. We will define a concrete Lie algebra to be a Lie algebra that is a subalgebra of for some vector space , and similarly define a representation of a Lie algebra to be a homomorphism into a concrete Lie algebra . It is a deep theorem of Ado (discussed in this previous post) that every abstract Lie algebra is in fact isomorphic to a concrete one (or equivalently, that every abstract Lie algebra has a faithful representation), but we will not need or prove this fact here.

Even without Ado’s theorem, though, the structure of abstract Lie algebras is very well understood. As with many other objects in an algebraic category, a basic way to understand a Lie algebra is to factor it into two simpler algebras via a short exact sequence

thus one has an injective homomorphism from to and a surjective homomorphism from to such that the image of the former homomorphism is the kernel of the latter. (To be pedantic, a short exact sequence in a general category requires these homomorphisms to be monomorphisms and epimorphisms respectively, but in the category of Lie algebras these turn out to reduce to the more familiar concepts of injectivity and surjectivity respectively.) Given such a sequence, one can (non-uniquely) identify with the vector space equipped with a Lie bracket of the form

for some bilinear maps and that obey some Jacobi-type identities which we will not record here. Understanding exactly what maps are possible here (up to coordinate change) can be a difficult task (and is one of the key objectives of Lie algebra cohomology), but in principle at least, the problem of understanding can be reduced to that of understanding that of its factors . To emphasise this, I will (perhaps idiosyncratically) express the existence of a short exact sequence (3) by the ATLAS-type notation

although one should caution that for given and , there can be multiple non-isomorphic that can form a short exact sequence with , so that is not a uniquely defined combination of and ; one could emphasise this by writing instead of , though we will not do so here. We will refer to as an extension of by , and read the notation (5) as “ is -by-“; confusingly, these two notations reverse the subject and object of “by”, but unfortunately both notations are well entrenched in the literature. We caution that the operation is not commutative, and it is only partly associative: every Lie algebra of the form is also of the form , but the converse is not true (see this previous blog post for some related discussion). As we are working in the infinitesimal world of Lie algebras (which have an additive group operation) rather than Lie groups (in which the group operation is usually written multiplicatively), it may help to think of as a (twisted) “sum” of and rather than a “product”; for instance, we have and , and also .

Special examples of extensions of by include the direct sum (or direct product) (also denoted ,) which is given by the construction (4) with and both vanishing, and the split extension (or semidirect product) (also denoted ), which is given by the construction (4) with vanishing and the bilinear map taking the form

for some representation of in the concrete Lie algebra of derivations of , that is to say the algebra of linear maps that obey the Leibniz rule

for all . (The derivation algebra of a Lie algebra is analogous to the automorphism group of a Lie group , with the two concepts being intertwined by the tangent space functor from Lie groups to Lie algebras (i.e. the derivation algebra is the infinitesimal version of the automorphism group). Of course, this functor also intertwines the Lie algebra and Lie group versions of most of the other concepts discussed here, such as extensions, semidirect products, etc.)

There are two general ways to factor a Lie algebra as an extension of a smaller Lie algebra by another smaller Lie algebra . One is to locate a Lie algebra ideal (or ideal for short) in , thus , where denotes the Lie algebra generated by , and then take to be the quotient space in the usual manner; one can check that , are also Lie algebras and that we do indeed have a short exact sequence

Conversely, whenever one has a factorisation , one can identify with an ideal in , and with the quotient of by .

The other general way to obtain such a factorisation is is to start with a homomorphism of into another Lie algebra , take to be the image of , and to be the kernel . Again, it is easy to see that this does indeed create a short exact sequence:

Conversely, whenever one has a factorisation , one can identify with the image of under some homomorphism, and with the kernel of that homomorphism. Note that if a representation is faithful (i.e. injective), then the kernel is trivial and is isomorphic to .

Now we consider some examples of factoring some class of Lie algebras into simpler Lie algebras. The easiest examples of Lie algebras to understand are the abelian Lie algebras , in which the Lie bracket identically vanishes. Every one-dimensional Lie algebra is automatically abelian, and thus isomorphic to the scalar algebra . Conversely, by using an arbitrary linear basis of , we see that an abelian Lie algebra is isomorphic to the direct sum of one-dimensional algebras. Thus, a Lie algebra is abelian if and only if it is isomorphic to the direct sum of finitely many copies of .

Now consider a Lie algebra that is not necessarily abelian. We then form the derived algebra ; this algebra is trivial if and only if is abelian. It is easy to see that is an ideal whenever are ideals, so in particular the derived algebra is an ideal and we thus have the short exact sequence

The algebra is the maximal abelian quotient of , and is known as the abelianisation of . If it is trivial, we call the Lie algebra perfect. If instead it is non-trivial, then the derived algebra has strictly smaller dimension than . From this, it is natural to associate two series to any Lie algebra , the lower central series

and the derived series

By induction we see that these are both decreasing series of ideals of , with the derived series being slightly smaller ( for all ). We say that a Lie algebra is nilpotent if its lower central series is eventually trivial, and solvable if its derived series eventually becomes trivial. Thus, abelian Lie algebras are nilpotent, and nilpotent Lie algebras are solvable, but the converses are not necessarily true. For instance, in the general linear group , which can be identified with the Lie algebra of complex matrices, the subalgebra of strictly upper triangular matrices is nilpotent (but not abelian for ), while the subalgebra of upper triangular matrices is solvable (but not nilpotent for ). It is also clear that any subalgebra of a nilpotent algebra is nilpotent, and similarly for solvable or abelian algebras.

From the above discussion we see that a Lie algebra is solvable if and only if it can be represented by a tower of abelian extensions, thus

for some abelian . Similarly, a Lie algebra is nilpotent if it is expressible as a tower of central extensions (so that in all the extensions in the above factorisation, is central in , where we say that is central in if ). We also see that an extension is solvable if and only of both factors are solvable.

For our next fundamental example of using short exact sequences to split a general Lie algebra into simpler objects, we observe that every abstract Lie algebra has an adjoint representation , where for each , is the linear map ; one easily verifies that this is indeed a representation (indeed, (2) is equivalent to the assertion that for all ). The kernel of this representation is the center , which the maximal central subalgebra of . We thus have the short exact sequence

which, among other things, shows that every abstract Lie algebra is a central extension of a concrete Lie algebra (which can serve as a cheap substitute for Ado’s theorem mentioned earlier).

For our next fundamental decomposition of Lie algebras, we need some more definitions. A Lie algebra is simple if it is non-abelian and has no ideals other than and ; thus simple Lie algebras cannot be factored into strictly smaller algebras . In particular, simple Lie algebras are automatically perfect and centerless. We have the following fundamental theorem:

Theorem 1 (Equivalent definitions of semisimplicity) Let be a Lie algebra. Then the following are equivalent:

• (i) does not contain any non-trivial solvable ideal.
• (ii) does not contain any non-trivial abelian ideal.
• (iii) The Killing form , defined as the bilinear form , is non-degenerate on .
• (iv) is isomorphic to the direct sum of finitely many non-abelian simple Lie algebras.

We review the proof of this theorem later in these notes. A Lie algebra obeying any (and hence all) of the properties (i)-(iv) is known as a semisimple Lie algebra. The statement (iv) is usually taken as the definition of semisimplicity; the equivalence of (iv) and (i) is then known as Weyl’s complete reducibility theorem, and the equivalence of (iv) and (iii) is known as the Cartan semisimplicity criterion. (The equivalence of (i) and (ii) is easy.)

If and are solvable ideals of a Lie algebra , then it is not difficult to see that the vector sum is also a solvable ideal (because on quotienting by we see that the derived series of must eventually fall inside , and thence must eventually become trivial by the solvability of ). As our Lie algebras are finite dimensional, we conclude that has a unique maximal solvable ideal, known as the radical of . The quotient is then a Lie algebra with trivial radical, and is thus semisimple by the above theorem, giving the Levi decomposition

expressing an arbitrary Lie algebra as an extension of a semisimple Lie algebra by a solvable algebra (and it is not hard to see that this is the only possible such extension up to isomorphism). Indeed, a deep theorem of Levi allows one to upgrade this decomposition to a split extension

although we will not need or prove this result here.

In view of the above decompositions, we see that we can factor any Lie algebra (using a suitable combination of direct sums and extensions) into a finite number of simple Lie algebras and the scalar algebra . In principle, this means that one can understand an arbitrary Lie algebra once one understands all the simple Lie algebras (which, being defined over , are somewhat confusingly referred to as simple complex Lie algebras in the literature). Amazingly, this latter class of algebras are completely classified:

Theorem 2 (Classification of simple Lie algebras) Up to isomorphism, every simple Lie algebra is of one of the following forms:

• for some .
• for some .
• for some .
• for some .
• , or .
• .
• .

(The precise definition of the classical Lie algebras and the exceptional Lie algebras will be recalled later.)

(One can extend the families of classical Lie algebras a little bit to smaller values of , but the resulting algebras are either isomorphic to other algebras on this list, or cease to be simple; see this previous post for further discussion.)

This classification is a basic starting point for the classification of many other related objects, including Lie algebras and Lie groups over more general fields (e.g. the reals ), as well as finite simple groups. Being so fundamental to the subject, this classification is covered in almost every basic textbook in Lie algebras, and I myself learned it many years ago in an honours undergraduate course back in Australia. The proof is rather lengthy, though, and I have always had difficulty keeping it straight in my head. So I have decided to write some notes on the classification in this blog post, aiming to be self-contained (though moving rapidly). There is no new material in this post, though; it is all drawn from standard reference texts (I relied particularly on Fulton and Harris’s text, which I highly recommend). In fact it seems remarkably hard to deviate from the standard routes given in the literature to the classification; I would be interested in knowing about other ways to reach the classification (or substeps in that classification) that are genuinely different from the orthodox route.

— 1. Abelian representations —

One of the key strategies in the classification of a Lie algebra is to work with representations of , particularly the adjoint representation , and then restrict such representations to various simpler subalgebras of , for which the representation theory is well understood. In particular, one aims to exploit the representation theory of abelian algebras (and to a lesser extent, nilpotent and solvable algebras), as well as the fundamental example of the two-dimensional special linear Lie algebra , which is the smallest and easiest to understand of the simple Lie algebras, and plays an absolutely crucial role in exploring and then classifying all the other simple Lie algebras.

We begin this program by recording the representation theory of abelian Lie algebras. We begin with representations of the one-dimensional algebra . Setting , this is essentially the representation theory of a single linear transformation . Here, the theory is given by the Jordan decomposition. Firstly, for each complex number , we can define the generalised eigenspace

One easily verifies that the are all linearly independent -invariant subspaces of , and in particular that there are only finitely many (the spectrum of ) for which is non-trivial. If one quotients out all the generalised eigenspaces, one can check that the quotiented transformation no longer has any spectrum, which contradicts the fundamental theorem of algebra applied to the characteristic polynomial of this quotiented transformation (or, if is more analytically inclined, one could apply Liouville’s theorem to the resolvent operators to obtain the required contradiction). Thus the generalised eigenspaces span :

On each space , the operator only has spectrum at zero, and thus (again from the fundamental theorem of algebra) has non-trivial kernel; similarly for any -invariant subspace of , such as the range of . Iterating this observation we conclude that is a nilpotent operator on , thus for some . If we then write to be the direct sum of the scalar multiplication operators on each generalised eigenspace , and to be the direct sum of the operators on these spaces, we have obtained the Jordan decomposition (or Jordan-Chevalley decomposition)

where the operator is semisimple in the sense that it is a diagonalisable linear transformation on (or equivalently, all generalised eigenspaces are actually eigenspaces), and is nilpotent. Furthermore, as we may use polynomial interpolation to find a polynomial such that vanishes to arbitrarily high order at for each (and also ), we see that (and hence ) can be expressed as polynomials in with zero constant coefficient; this fact will be important later. In particular, and commute.

Conversely, given an arbitrary linear transformation , the Jordan-Chevalley decomposition is the unique decomposition into commuting semisimple and nilpotent elements. Indeed, if we have an alternate decomposition into a semisimple element commuting with a nilpotent element , then the generalised eigenspaces of must be preserved by both and , and so without loss of generality we may assume that there is just a single generalised eigenspace ; subtracting we may then assume that , but then is nilpotent, and so is also nilpotent; but the only transformation which is both semisimple and nilpotent is the zero transformation, and the claim follows.

From the Jordan-Chevalley decomposition it is not difficult to then place in Jordan normal form by selecting a suitable basis for ; see e.g. this previous blog post. But in contrast to the Jordan-Chevalley decomposition, the basis is not unique in general, and we will not explicitly use the Jordan normal form in the rest of this post.

Given an abstract complex vector space , there is in general no canonical notion of complex conjugation on , or of linear transformations . However, we can define the conjugate of any semisimple transformation , defined as the direct sum of on each eigenspace of . In particular, we can define the conjugate of the semisimple component of an arbitrary linear transformation , which will be the direct sum of on each generalised eigenspace of . The significance of this transformation lies in the observation that the product has trace on each generalised eigenspace (since nilpotent operators have zero trace), and in particular we see that

if and only if the spectrum consists only of zero, or equivalently that is nilpotent. Thus (7) provides a test for nilpotency, which will be turn out to be quite useful later in this post. (Note that this trick relies very much on the special structure of , in particular the fact that it has characteristic zero.)

In the above arguments we have used the basic fact that if two operators and commute, then the generalised eigenspaces of one operator are preserved by the other. Iterating this fact, we can now start understanding the representations of an abelian Lie algebra. Namely, there is a finite set of linear functionals (or homomorphisms) on (i.e. elements of the dual space ) for which the generalised eigenspaces

are non-trivial and -invariant, and we have the decomposition

Here we use as short-hand for writing for all . An important special case arises when the action of is semisimple in the sense that is semisimple for all . Then all the generalised eigenspaces are just eigenspaces (or weight spaces) , thus

for all and . When this occurs we call a weight vector with weight .

— 2. Engel’s theorem and Lie’s theorem —

In the introduction we gave the two basic examples of nilpotent and solvable Lie algebras, namely the strictly upper triangular and upper triangular matrices. The theorems of Engel and of Lie assert, roughly speaking, that these examples (and subalgebras thereof) are essentially the only type of solvable and nilpotent Lie algebras that can exist, at least in the concrete setting of subalgebras of . Among other things, these theorems greatly clarify the representation theory of nilpotent and solvable Lie algebras.

We begin with Engel’s theorem.

Theorem 3 (Engel’s theorem) Let be a concrete Lie algebra such that every element of is nilpotent as a linear transformation on .

• (i) If is non-trivial, then there is a non-zero element of which is annihilated by every element of .
• (ii) There is a basis of for which all elements of are strictly upper triangular. In particular, is nilpotent.

Proof: We begin with (i). We induct on the dimension of . The claim is trivial for dimensions and , so suppose that has dimension greater than , and that the claim is already proven for smaller dimensions.

Let be a maximal proper subalgebra of , then has dimension strictly between zero and (since all one-dimensional subspaces are proper subalgebras). Observe that for every , acts on both the vector spaces and and thus also on the quotient space . As is nilpotent, all of these actions are nilpotent also. In particular, by induction hypothesis, there is which is annihilated by for all . Let be a representative of in , then , and so is a subalgebra and is thus all of .

By induction hypothesis again, the space of vectors in annihilated by is non-trivial; as , it is preserved by . As is nilpotent, there is a non-trivial element of annihilated by and hence by , as required.

Now we prove (ii). We induct on the dimension of . The case of dimension zero is trivial, so suppose has dimension at least one, and the claim has already been proven for dimension . By (i), we may find a non-trivial vector annihilated by , and so we may project down to . By the induction hypothesis, there is a basis for on which the projection of any element of is strictly upper-triangular; pulling this basis back to and adjoining , we obtain the claim.

As a corollary of this theorem and the short exact sequence (6) we see that an abstract Lie algebra is nilpotent iff is nilpotent iff is nilpotent in for every (i.e. every element of is ad-nilpotent).

Engel’s theorem is in fact valid over every field. The analogous theorem of Lie for solvable algebras, however, relies much more strongly on the specific properties of the complex field .

Theorem 4 (Lie’s theorem) Let be a solvable concrete Lie algebra.

• (i) If is non-trivial, there exists a non-zero element of which is an eigenvector for every element of .
• (ii) There is a basis for such that every element of is upper triangular.

Note that if one specialises Lie’s theorem to abelian then one essentially recovers the abelian theory of the previous section.

Proof: We prove (i). As before we induct on the dimension of . The dimension zero case is trivial, so suppose that has dimension at least one and that the claim has been proven for smaller dimensions.

Let be a codimension one subalgebra of ; such an algebra can be formed by taking a codimension one subspace of the abelianisation (which has dimension at least one, else will not be solvable) and then pulling back to . Note that is automatically an ideal.

By induction, there is a non-zero element of such that every element of has as an eigenvector, thus we have

for all and some linear functional . If we then set to be the simultaneous eigenspace

then is a non-trivial subspace of .

Let be an element of that is not in , and let . Consider the space spanned by the orbit . By finite dimensionality, this space has a basis for some . By induction and definition of , we see that every acts on this space by an upper-triangular matrix with diagonal entries in this basis. Of course, acts on this space as well, and so has trace zero on this space, thus and so (here we use the characteristic zero nature of ). From this we see that fixes . If we let be an eigenvector of on (which exists from the Jordan decomposition of ), we conclude that is a simultaneous eigenvector of as required.

The claim (ii) follows from (i) much as in Engel’s theorem.

— 3. Characterising semisimplicity —

The objective of this section will be to prove Theorem 1.

Let be an concrete Lie algebra, and be an element of . Then the components of need not lie in . However they behave “as if” they lie in for the purposes of taking Lie brackets, in the following sense:

Lemma 5 Let and let have Jordan decomposition . Then , and .

Proof: As and are semisimple and nilpotent on and commute with each other, and are semisimple and nilpotent on and also commute with each other (this can for instance by using Lie’s theorem (or the Jordan normal form) to place in upper triangular form and computing everything explicitly). Thus is the Jordan-Chevalley decomposition of , and in particular for some polynomial with zero constant coefficient. Since maps to the subalgebra , we conclude that does also, thus as required. Similarly for and (note that ).

We can now use this (together with Engel’s theorem and the test (7) for nilpotency) to obtain a part of Theorem 1:

Proposition 6 Let be a simple Lie algebra. Then the Killing form is non-degenerate.

Proof: As is simple, its center is trivial, so by (6) is isomorphic to . In particular we may assume that is a concrete Lie algebra, thus for some vector space .

Suppose for contradiction that is degenerate. Using the skew-adjointness identity

for all (which comes from the cyclic properties of trace), we see that the kernel is a non-trivial ideal of , and is thus all of as is simple. Thus for all .

Now let . By Lemma 5, acts by Lie bracket on and so one can define . We now consider the quantity

We can rearrange this as

By Lemma 5, , so this is equal to

and so

for all . On the other hand, is an ideal of ; as is simple, we must thus have (i.e. is perfect). As , we conclude that

From (7) we conclude that is nilpotent for every . By Engel’s theorem, this implies that , and hence , is nilpotent; but is simple, giving the desired contradiction.

Corollary 7 Let be a simple ideal of a Lie algebra . Then is complemented by another ideal of (thus and ), with isomorphic to the direct sum .

Proof: The adjoint action of restricts to the ideal and gives a restricted Killing form

By Proposition 6, this bilinear form is non-degenerate on , so the orthogonal complement

is a complementary subspace to . It can be verified to also be an ideal. Since lies in both and , we see that , and so is isomorphic to as claimed.

Now we can prove Theorem 1. We first observe that (i) trivially implies (ii); conversely, if has a non-trivial solvable ideal , then every element of the derived series of is also an ideal of , and in particular will have a non-trivial abelian ideal. Thus (i) and (ii) are equivalent.

Now we show that (i) implies (iv), which we do by induction on the dimension of . Of course we may assume is non-trivial. Let be a non-trivial ideal of of minimal dimension. If then is simple (note that it cannot be abelian as is non-trivial and semisimple) and we are done. If is strictly smaller than , then it also has no non-trivial solvable ideals (because the radical of is a characteristic subalgebra of and is thus an ideal in ) and so by induction is isomorphic to the direct sum of simple Lie algebras; as was minimal, we conclude that is itself simple. By Corollary 7, then splits as the direct sum of and a semisimple Lie algebra of strictly smaller dimension, and the claim follows from the induction hypothesis.

From Proposition 6 we see that (iv) implies (iii), so to finish the proof of Theorem 1 it suffices to show that (iii) implies (ii). Indeed, if has a non-trivial abelian ideal , then for any and , annihilates and also has range in , hence has trace zero, so is -orthogonal to , giving the degeneracy of the Killing form.

Remark 1 Similar methods also give the Cartan solvability criterion: a Lie algebra is solvable if and only if is orthogonal to with respect to the Killing form. Indeed, the “only if” part follows easily from Lie’s theorem, while for the “if” part one can adapt the proof of Proposition 6 to show that if is orthogonal to , then every element of is nilpotent, hence by Engel’s theorem is nilpotent, and so from the short exact sequence (6) we see that is nilpotent, and hence is solvable.

Remark 2 The decomposition of a semisimple Lie algebra as the direct sum of simple Lie algebras is unique up to isomorphism and permutation. Indeed, suppose that is isomorphic to for some simple . We project each to and observe from simplicity that these projections must either be zero or isomorphisms (cf. Schur’s lemma). For fixed , there must be at least one for which the projection is an isomorphism (otherwise could not generate all of ); on the other hand, as any two commute with each other in the direct sum, and is nonabelian, there is at most one for which the projection is an isomorphism. This gives the required identification of the and up to isomorphism and permutation.

Remark 3 One can also establish complete reducibility by using the Weyl unitary trick, in which one first creates a real compact Lie group whose Lie algebra is a real form of the complex Lie algebra being studied, and then uses the complete reducibility of actions of compact groups.

Semisimple Lie algebras have a number of important non-degeneracy properties. For instance, they have no non-trivial outer automorphisms (at the infinitesimal level, at least):

Lemma 8 Let be a semisimple Lie algebra. Then every derivation on is inner, thus for some .

Proof: From the identity we see that is an ideal in . The trace form on restricts to the Killing form on , which is non-degenerate.

Suppose for contradiction that is not all of , then there is a non-trivial derivation which is trace-form orthogonal to , thus is trace-orthogonal to for all , so that is trace-orthogonal to for all . As is non-degenerate, we conclude that for all , and so is trivial, a contradiction.

This implies that the Jordan decomposition preserves concrete semisimple Lie algebras:

Corollary 9 Let be a concrete semisimple Lie algebra, and let . Then also lie in .

Proof: By Lemma 7, the operation is a derivation on , thus there exists such that for all , thus centralises . In particular, preserves all the generalised eigenspaces of . On each such eigenspace , every element of has trace zero since ; in particular, and have trace zero, and so has trace zero on also. But the trace of on the eigenspace is just , so we conclude that is trivial and so . Similarly for and .

This allows us to make the Jordan decomposition universal for semisimple algebras:

Lemma 10 (Semisimple Jordan decomposition) Let be a semisimple Lie algebra, and let . Then we have a unique decomposition in such that and for every representation of .

Proof: As the adjoint representation is faithful we may assume without loss of generality that is a concrete algebra, thus . The uniqueness is then clear by taking to be the identity. To obtain existence, we take to be the concrete Jordan decomposition. We need to verify and for any representation . The adjoint actions of and on commute and are semisimple and nilpotent respectively and so

in (cf. the proof of Lemma 5). A similar argument (applying Corollary 9 to , which is isomorphic to a quotient of and is thus semisimple, to keep in ) gives

Since the adjoint representation of the semisimple algebra is faithful, the claim follows.

One can also show that , commute with each other and with the centraliser of by using the faithful nature of the adjoint representation for semisimple algebras, though we will not need these facts here. Using this lemma we have a well-defined notion of an element of a semisimple algebra being semisimple (resp. nilpotent), namely that or . Lemma 10 then implies that any representation of a semisimple element of is again semisimple, and any representation of a nilpotent element of is again nilpotent. This apparently innocuous statement relies heavily on the semisimple nature of ; note for instance that the representation

of the non-semisimple algebra into takes semisimple elements to nilpotent ones.

— 4. Cartan subalgebras —

While simple Lie algebras do not have any non-trivial ideals, they do have some very useful subalgebras known as Cartan subalgebras which will eventually turn out to be abelian and which can be used to dramatically clarify the structure of the rest of the algebra.

We need some definitions. An element of is said to be regular if its generalised null space

has minimal dimension. A Cartan subalgebra of is a nilpotent subalgebra of which is its own normaliser, thus is equal to . From the polynomial nature of the Lie algebra operations (and the Noetherian nature of algebraic geometry) we see that the regular elements of are generic (i.e. they form a non-empty Zariski-open subset of ).

Example 1 In , the regular elements consist of the semisimple elements with distinct eigenvalues. Fixing a basis for , the space of elements of that are diagonalised by that basis form a Cartan subalgebra of .

Cartan algebras always exist, and can be constructed as generalised null spaces of regular elements:

Proposition 11 (Existence of Cartan subalgebras) Let be an abstract Lie algebra. If is regular, then the generalised null space of is a Cartan subalgebra.

Proof: Suppose that is not nilpotent, then by Engel’s theorem the adjoint action of at least one element of on is not nilpotent. By the polynomial nature of the Lie algebra operations, we conclude that the adjoint action of a generic element of on is not nilpotent.

The action of on is non-singular, so the action of generic elements of on is also non-singular. Thus we can find such that is not nilpotent on and not singular on . From this we see that is a proper subspace of , contradicting the regularity of . Thus is nilpotent.

Finally, we show that is its own normaliser. Suppose that normalises , then . But is the generalised null space of , and so as required.

Furthermore, all Cartan algebras arise as generalised null spaces:

Proposition 12 (Cartans are null spaces) Let be an abstract Lie algebra, and let be a Cartan subalgebra. Let

be the generalised null space of . Then . Furthermore, for generic , one has

Proof: As is nilpotent, we certainly have . Now, for any , acts nilpotently on both and and hence on . By Engel’s theorem, we can thus find that is annihilated by the adjoint action of ; pulling back to , we conclude that the normaliser of is strictly larger than , contradicting the hypothesis that is a Cartan subalgebra. This shows that .

Now let be generic, then has minimal dimension amongst . Let be arbitrary. Then for any scalar , acts on and on and hence on . This action is invertible when , and hence is also invertible for generic ; thus for generic , . By minimality we conclude that , so is nilpotent on for generic , and thus for all . In particular is nilpotent on for any , thus . Since , we obtain as required.

Corollary 13 (Cartans are conjugate) Let be a Lie algebra, and let be a Cartan algebra. Then for generic , is conjugate to by an inner automorphism of (i.e. an element of the algebraic group generated by for ). In particular, any two Cartan subalgebras are conjugate to each other by an inner automorphism.

Proof: Let be the set of with , then is a Zariski open dense subset of by Proposition 12. Then let be the collection of that are conjugate to an , then is a algebraically constructible subset of . For , observe that and span , since , and so by the inverse function theorem, a (topological) neighbourhood of is contained in . This implies that is Zariski dense, and the claim follows.

In the case of semisimple algebras, the Cartan structure is particularly clean:

Proposition 14 Let be a semisimple Lie algebra. Then every Cartan subalgebra is abelian, and is non-degenerate on .

The dimension of the Cartan algebra of a semisimple Lie algebra is known as the rank of the algebra.

Proof: The nilpotent algebra acts via the adjoint action on , and by Lie’s theorem this action can be made upper triangular. From this it is not difficult to obtain a decomposition

for some finite set , where are the generalised eigenspaces

From the Jacobi identity (2) we see that . Among other things, this shows that has ad-trace zero for any non-zero , and hence are -orthogonal if . In particular, is -orthogonal to . By Theorem 1, is non-degenerate on , and thus also non-degenerate on ; by Proposition 12, is thus non-degenerate on . But by Lie’s theorem, we can find a basis for which consists of upper-triangular matrices in the adjoint representation of , so that is strictly upper-triangular and thus -orthogonal to . As is non-degenerate on , this forces to be abelian, as required.

We now use the semisimple Jordan decomposition (Lemma 10) to obtain a further non-degeneracy property of the Cartan subalgebras of semisimple algebras:

Proposition 15 Let be a semisimple Lie algebra. Then every Cartan subalgebra consists entirely of semisimple elements.

Proof: Let , then (by the abelian nature of ) annihilates ; as is a polynomial in with zero constant coefficient, annihilates as well; thus normalises and thus also lies in as is Cartan. If , then commutes with and so commutes with . As the latter is nilpotent, we conclude that is nilpotent and thus has trace zero. Thus is -orthogonal to and thus vanishes since the Killing form is non-degenerate on . Thus every element of is semisimple as required.

— 5. representations —

To proceed further, we now need to perform some computations on a very specific Lie algebra, the special linear algebra of complex matrices with zero trace. This is a three-dimensional concrete Lie algebra, spanned by the three generators

which obey the commutation relations

Conversely, any abstract three-dimensional Lie algebra generated by with relations (8) is clearly isomorphic to . One can check that this is a simple Lie algebra, with the one-dimensional space generated by being a Cartan subalgebra.

Now we classify by hand the representations of . Observe that acts infinitesimally on by the differential operators (or vector fields)

In particular, we see that for each natural number , the space of homogeneous polynomials in two variables of degree has a representation ; if we give this space the basis for , the action is then described by the formulae

for . From these formulae it is also easy to see that these representations are irreducible in the sense that the have no non-trivial -invariant subspaces.

Conversely, these representations (and their direct sums) describe (up to isomorphism) all of the representations of :

Theorem 16 (Representations of ) Any representation is isomorphic to the direct sum of finitely many of the representations .

Here of course the direct sum of two representations , is defined as , and two representations , are isomorphic if there is an invertible linear map such that for all .

Proof: By induction we may assume that is non-trivial, the claim has already been proven for any smaller dimensional spaces than .

As is semisimple, is semisimple by Lemma 10, and so we can split into the direct sum

of eigenspaces of for some finite .

From (8) we have the raising law

and the lowering law

As is finite, we may find a “highest weight” with the property that , thus annihilates by the raising law. We will use the basic strategy of starting from the highest weight space and applying lowering operators to discover one of the irreducible components of the representation.

From (8) one has

and so from induction and the lowering law we see that

for all natural numbers and all . If is never zero, this creates an infinite sequence of non-trivial eigenspaces, which is absurd, so we have for some natural number , thus . If we then let

then we see that is invariant under , , and , and thus -invariant; also if for each we let be the set of all such that is never a non-zero element of then we see that

is also -invariant, and furthermore that and are complementary subspaces in . Applying the induction hypothesis, we are done unless , but then by splitting into one-dimensional spaces and applying the lowering operators, we see that we reduce to the case that is one-dimensional. But if one then lets be a generator of and recursively defines by

one then checks using (10) that is isomorphic to , and the claim follows.

Remark 4 Theorem 16 shows that all representations of are completely reducible in that they can be decomposed as the direct sum of irreducible representations. In fact, all representations of semisimple Lie algebras are completely reducible; this can be proven by a variant of the above arguments (in combination with the analysis of weights given below), and can also be proven by the unitary trick, or by analysing the action of Casimir elements of the universal enveloping algebra of . See e.g. Fulton-Harris for details.

— 6. Root spaces —

Now we use the theory to analyse more general semisimple algebras.

Let be a semisimple Lie algebra, and let be a Cartan algebra, then by Proposition 14 is abelian and acts in a semisimple fashion on , and by Proposition 12 is its own null space in the weight decomposition of , thus we have the Cartan decomposition

as vector spaces (not as Lie algebras) where is a finite subset of (known as the set of roots) and is the non-trivial eigenspace

Example 2 A key example to keep in mind is when is the Lie algebra of matrices of trace zero. An explicit computation using the Killing form and Theorem 1 shows that this algebra is semisimple; in fact it is simple, but we will not show this yet. The space of diagonal matrices of trace zero can then be verified to be a Cartan algebra; it can be identified with the space of complex -tuples summing to zero, and using the usual Hermitian inner product on we can also identify with . The roots are then of the form for distinct , where is the standard basis for , with being the one-dimensional space of matrices that are vanishing except possibly at the coefficient.

From the Jacobi identity (2) we see that the Lie bracket acts additively on the weights, thus

for all . Taking traces, we conclude that

whenever . As is non-degenerate, we conclude that if is non-trivial, then must also be non-trivial, thus is symmetric around the origin.

We also claim that spans as a vector space. For if this were not the case, then there would be a non-trivial that is annihilated by , which by (11) implies that annihilates all of the and is thus central, contradicting the semisimplicity of .

From Proposition 14, is non-degenerate on . Thus, for each root , there is a corresponding non-zero element of such that for all . If we let , and , we have

and thus by the non-degeneracy of on we obtain the useful formula

for and .

As is non-degenerate, we can find and with (which can be found as is non-degenerate). We divide into two cases depending on whether vanishes or not. If vanishes, then is non-trivial but commutes with and , and so generate a solvable algebra. By Lie’s theorem, this algebra is upper-triangular in some basis, and so is nilpotent, hence is nilpotent; but by Proposition 15 is also semisimple, contradicting the non-zero nature of (and the semisimple nature of ). Thus is non-vanishing. If we then scale so that , where is the co-root of , defined as the element of given by the formula

so that

then obey the relations (8) and thus generate a copy of , rather than a solvable algebra. The representation theory of can then be applied to the space

where . By (19), this space is invariant with respect to and and hence to the copy of , and by (11), (14) each is the weight space of of weight for each . By Theorem 16, we conclude that the set consists of integers. On the other hand, from (13) we see that any copy of the representation with a positive even integer must have its weight space contained in the span of , and so there is only one such representation in (15). As already give a copy of in (15), there are no other copies of with positive even, thus we have that is one-dimensional and that the only even multiples of in are . In particular, whenever , which also implies that whenever . Returning to Theorem 16, we conclude that the set contains no odd integers, and so and are the only multiples of in .

Next, let be any non-zero element of orthogonal to with respect to the inner product of that is dual to the restriction of the Killing form to , and consider the space

where

By (19), this is again an -invariant space, and by (11), (14) each is the weight space of of weight . From Theorem 16 we see that is an arithmetic progression of spacing ; in particular, is symmetric around the origin and consists only of integers. This implies that the set is symmetric with respect to reflection across the hyperplane that is orthogonal to , and also implies that

for all roots .

We summarise the various geometric properties of as follows:

Proposition 17 (Root systems) Let be a semisimple Lie algebra, let be a Cartan subalgebra, and let be the inner product on that is dual to the Killing form restricted to . Let be the set of roots. Then:

• (i) does not contain zero.
• (ii) If is a root, then is symmetric with respect to the reflection operation across the hyperplane orthogonal to ; in particular, is also a root.
• (iii) If is a root, then no multiple of other than are roots.
• (iv) If are roots, then is an integer or half-integer. Equivalently, for some integer .
• (v) spans .

A set of vectors obeying the above axioms (i)-(v) is known as a root system on (viewed as a finite dimensional complex Hilbert space with the inner product ).

Remark 5 A short calculation reveals the remarkable fact that if is a root system, then the associated system of co-roots is also a root system. This is one of the starting points for the deep phenomenon of Langlands duality, which we will not discuss here.

When is simple, one can impose a useful additional axiom on . Say that a root system is irreducible if cannot be covered by the union of two orthogonal proper subspaces of .

Lemma 18 If is a simple Lie algebra, then the root system of is irreducible.

Proof: If can be covered by two orthogonal subspaces , then if we consider the subspace of

where we use the inner product to identify with and thus with a subspace of (thus for instance this identifies with ), then one can check using (19) and (13) that this is a proper ideal of , contradicting simplicity.

It is easy to see that every root system is expressible as the union of irreducible root systems (on orthogonal subspaces of ). As it turns out, the irreducible root systems are completely classified, with the complete list of root systems (up to isomorphism) being described in terms of the Dynkin diagrams briefly mentioned in Theorem 2. We will now turn to this classification in the next section, and then use root systems to recover the Lie algebra.

— 7. Classification of root systems —

In this section we classify all the irreducible root systems on a finite dimensional complex Hilbert space , up to Hilbert space isometry. Of course, we may take to be a standard complex Hilbert space without loss of generality. The arguments here are purely elementary, proceeding purely from the root system axioms rather than from any Lie algebra theory.

Actually, we can quickly pass from the complex setting to the real setting. By axiom (v), contains a basis of ; by axiom (iv), the inner products between these basis vectors are real, as are the inner products between any other root and a basis root. From this we see that lies in the real vector space spanned by the basis roots, so by a change of basis we may assume without loss of generality that .

Henceforth is assumed to lie in . From two applications of (iv) we see that for any two roots , the expression

lies in ; but it is also equal to , and hence

for all roots . Analysing these cases further using (iv) again, we conclude that there are only a restricted range of options for a pair of roots :

Lemma 19 Let be roots. Then one of the following occurs:

• (0) and are orthogonal.
• (1/4) have the same length and subtend an angle of or .
• (1/2) has times the length of or vice versa, and subtend an angle of or .
• (3/4) has times the length of or vice versa, and subtend an angle of or .
• (1) .

We next record a useful corollary of Lemma 19 (and axiom (ii)):

Corollary 20 Let be roots. If subtend an acute angle, then and are also roots, but is not a root. Equivalently, if subtend an obtuse angle, then is a root, but and are not roots.

This follows from a routine case analysis and is omitted.

We can leverage Corollary 20 as follows. Call an element of regular if it is not orthogonal to any root, thus generic elements of are regular. Given a regular element , let denote the roots which are -positive in the sense that their inner product with is positive; thus is partitioned into and . We will abbreviate -positive as positive if is understood from context. Call a positive root a -simple root (or simple root for short) if it cannot be written as the sum of two positive roots. Clearly every positive root is then a linear combination of simple roots with natural number coefficients. By Corollary 20, two simple roots cannot subtend an acute angle, and so any two distinct simple roots subtend a right or obtuse angle.

Example 3 Using the root system of discussed previously, if one takes to be any vector in with decreasing coefficients, then the positive roots are those roots with , and the simple roots are the roots for .

Define an admissible configuration to be a collection of unit vectors in in a open half-space with the property that any two vectors in this collection form an angle of , , , or , and call the configuration irreducible if it cannot be decomposed into two non-empty orthogonal subsets. From Lemma 19 and the above discussion we see that the unit vectors associated to the simple roots are an admissible configuration. They are also irreducible, for if the simple roots partition into two orthogonal sets then it is not hard to show (using Corollary 20) that all positive roots lie in the span of one of these two sets, contradicting irreducibility of the root system.

We can say quite a bit about admissible configurations; the fact that the vectors in the system always subtend right or obtuse angles, combined with the half-space restriction, is quite limiting (basically because this information can be in violation of inequalities such as the Bessel inequality, or the positive (semi-)definiteness of the Gram matrix). We begin with an assertion of linear independence:

Lemma 21 If is an admissible configuration, then it is linearly independent.

Among other things, this shows that the number of simple roots of a semisimple Lie algebra is equal to the rank of that algebra.

Proof: Suppose this is not the case, then one has a non-trivial linear constraint

for some positive and disjoint . But as any two vectors in an admissible configuration subtend a right or obtuse angle, , and thus . But this is not possible as all the lie in an open half-space.

Define the Coxeter diagram of an admissible configuration to be the graph with vertices , and with any two vertices connected by an edge of multiplicity , thus two vertices are unconnected if they are orthogonal, connected with a single edge if they subtend an angle of , a double edge if they subtend an angle of , and a triple edge if they subtend an angle of . The irreducibility of a configuration is equivalent to the connectedness of a Coxeter diagram. Note that the Coxeter diagram describes all the inner products between the and thus describes the up to an orthogonal transformation (as can be seen for instance by applying the Gram-Schmidt process).

Lemma 22 The Coxeter diagram of an admissible configuration is acyclic (ignoring multiplicity of edges). In particular, the Coxeter diagram of an irreducible admissible configuration is a tree.

Proof: Suppose for contradiction that the Coxeter diagram contains a cycle , we see that for (with the convention ) and for all other . This implies that , which contradicts the linear independence of the .

Lemma 23 Any vertex in the Coxeter diagram has degree at most three (counting multiplicity).

Proof: Let be a vertex which is adjacent to some other vertices , which are then an orthonormal system. By Bessel’s inequality (and linear independence) one has

But from construction of the Coxeter diagram we have for each , where is the multiplicity of the edge connecting and . The claim follows.

We can also contract simple edges:

Lemma 24 If is an admissible configuration with joined by a single edge, then the configuration formed from by replacing with the single vertex is again an admissible configuration, with the resulting Coxeter diagram formed from the original Coxeter diagram by deleting the edge between and and then identifying together.

This follows easily from acyclicity and direct computation.

By Lemma 23 and Lemma 24, the Coxeter diagram can never form a vertex of degree three no matter how many simple edges are contracted. From this we can easily show that connected Coxeter diagrams must have one of the following shapes:

• : vertices joined in a chain of simple edges;
• : vertices joined in a chain of edges, one of which is a double edge and all others are simple edges;
• : three chains of simple edges emenating from a common vertex (forming a “Y” shape), connecting vertices in all;
• : Two vertices joined by a triple edge.

We can cut down the and cases further:

Lemma 25 The Coxeter diagram of an admissible configuration cannot contain as a subgraph

• (a) A chain of four edges, with one of the interior edges a double edge;
• (b) Three chains of two simple edges each, emenating from a common vertex;
• (c) Three chains of simple edges of length respectively, emenating from a common vertex.

Proof: To exclude (a), suppose for contradiction that we have two chains and of simple edges, with joined by a double edge. Writing and , one computes that are unit vectors with inner product , implying that are parallel, contradicting linear independence.

To exclude (b), suppose that we have three chains , , of simple edges joined at . Then the vectors are an orthonormal system that each have an inner product of each with . Comparing this with Bessel’s inequality we conclude that lies in the span of , contradicting linear independence.

Finally, to exclude (c), suppose we have three chains , , of simple edges joined at . Writing , , , we compute that are an orthonormal system that have inner products of respectively with . As , this forces to lie in the span of , again contradicting linear independence.

We remark that one could also obtain the required contradictions in the above proof by verifying in all three cases that the Gram matrix of the subconfiguration has determinant zero.

Corollary 26 The Coxeter diagram of an irreducible admissible configuration must take one of the following forms:

• : vertices joined in a chain of simple edges for some ;
• : vertices joined in a chain of edges for some , with one boundary edge being a double edge and all other edges simple;
• : Three chains of simple edges of length respectively for some , emenating from a single vertex;
• : Three chains of simple edges of length respectively for some , emenating from a single vertex;
• : Four vertices joined in a chain of edges, with the middle edge being a double edge and the other two edges simple;
• : Two vertices joined by a triple edge.

Now we return to root systems. Fixing a regular , we define the Dynkin diagram to be the Coxeter diagram associated to the (unit vectors of the) simple roots, except that we orient the double or triple edges to point from the longer root to the shorter root. (Note from Lemma 19 that we know exactly what the ratio between lengths is in these cases; in particular, the Dynkin diagram describes the root system up to a unitary transformation and dilation.) We conclude

Corollary 27 The Dynkin diagram of an irreducible root system must take one of the following forms:

• : vertices joined in a chain of simple edges for some ;
• : vertices joined in a chain of edges for some , with one boundary edge being a double edge (pointing outward) and all other edges simple;
• : vertices joined in a chain of edges for some , with one boundary edge being a double edge (pointing inward) and all other edges simple;
• : Three chains of simple edges of length respectively for some , emenating from a single vertex;
• : Three chains of simple edges of length respectively for some , emenating from a single vertex;
• : Four vertices joined in a chain of edges, with the middle edge being a double (oriented) edge and the other two edges simple;
• : Two vertices joined by a triple (oriented) edge.

This describes (up to isomorphism and dilation) the simple roots:

• : The simple roots take the form for in the space of vectors whose coefficients sum to zero;
• : The simple roots take the form for and also in .
• : The simple roots take the form for and also in .
• : The simple roots take the form for and also in .
• : The simple roots take the form for and also and in .
• : This system is obtained from by deleting the first one or two simple roots (and cutting down appropriately)
• : The simple roots take the form for and also and in .
• : The simple roots take the form , in .

Now we show how the simple roots can be used to recover the entire root system. Define the Weyl group to be the group generated by all the reflections coming from all the roots ; as the roots span and obey axiom (ii), the Weyl group acts faithfully on the finite set and is thus itself finite.

Lemma 28 Let be regular, and let be any element of . Then there exists such that for all -simple roots (or equivalently, for all -positive roots ). In particular, if is regular, then , so that all -simple roots are -simple and vice versa.

Furthermore, every root can be mapped by an element of to an -simple root.

Finally, is generated by the reflections coming from the -simple roots .

Proof: Let be a simple root. The action of the reflection maps to , and maps all other simple roots to for some non-negative (since subtend a right or obtuse angle). In particular, we see that maps all positive roots other than to positive roots, and hence (as is an involution)

In particular, if we define , then

for all simple roots .

Let be the subgroup of generated by the for the simple roots , and choose to maximise . Then from (17) we have , giving the first claim. Since every root is -simple for some regular (by selecting to very nearly be orthogonal to ), we conclude that every root can be mapped by an element of to a -simple root in , giving the second claim. Thus for any root , is conjugate in to a reflection for a -simple root , so lies in and so , giving the final claim.

Remark 6 The set of all for which is known as the Weyl chamber associated to ; this is an open polyhedral cone in , and the above lemma shows that it is the interior of a fundamental domain of the action of the Weyl group. In the case of the special linear group, the standard Weyl chamber (in now instead of ) would be the set of vectors with decreasing coefficients.

From the above lemma we can reconstruct the root system from the simple roots by using the reflections associated to the simple roots to generate the Weyl group , and then applying the Weyl group to the simple roots to recover all the roots. Note that the lemma also shows that the set of -simple roots and -simple roots are isomorphic for any regular , so that the Dynkin diagram is indeed independent (up to isomorphism) of the choice of regular element as claimed earlier. We have thus in principle described the irreducible root systems (up to isomorphism) as coming from the Dynkin diagrams ; see for instance the Wikipedia page on root systems for explicit descriptions of all of these. With these explicit descriptions one can verify that all of these systems are indeed irreducible root systems.

— 8. Chevalley bases —

Now that we have described root systems, we use them to reconstruct Lie algebras. We first begin with an abstract uniqueness result that shows that a simple Lie algebra is determined up to isomorphism by its root system.

Theorem 29 (Root system uniquely determines a simple Lie algebra) Let be simple Lie algebras with Cartan subalgebras , and root systems , . Suppose that one can identify with as vector spaces in such a way that the root systems agree: . Then the identification between and can be extended to an identification of and as Lie algebras.

Proof: First we note from (11) and the identification that the Killing forms on and agree, so we will identify as Hilbert spaces, not just as vector spaces.

The strategy will be exploit a Lie algebra version of the Goursat lemma (or the Schur lemma), finding a sufficiently “non-degenerate” subalgebra of and using the simple nature of and to show that this subalgebra is the graph of an isomorphism from to . This strategy will follow the same general strategy used in Theorem 16, namely to start with a “highest weight” space and apply lowering operators to discover the required graph.

We turn to the details. Pick a regular element of , so that one has a notion of a positive root. For every simple root , we select non-zero elements , of respectively such that

where is the co-root of ; similarly select in , and set and . Let be the subalgebra of generated by the and . It is not hard to see that the generate as a Lie algebra, so surjects onto ; similarly surjects onto .

Let be a maximal root, that is to say a root such that is not a root for any positive ; such a root always exists. (It is in fact unique, though we will not need this fact here.) Then we have one-dimensional spaces and , and thus a two-dimensional subspace in . Inside this subspace, we select a one-dimensional subspace which is not equal to or ; in particular, is not contained in or .

Let be the subspace of generated by and the adjoint action of the lowering operators , thus it is spanned by elements of the form

for simple roots and . Then contains and is thus not contained in ; because (19) only involves lowering operators, we also see that does not contain any other element of other than . In particular, is not all of .

Clearly is closed under the adjoint action of the lowering operators . We claim that it is also closed under the adjoint action of the raising operators . To see this, first observe that commute when are distinct simple roots, because cannot be a root (since this would make one of non-simple). Next, from (18) we see that acts as a scalar on any element of the form (19), while from the maximality of we see that annihilates . From this the claim easily follows.

As is closed under the adjoint action of both the and the , we have . Projecting onto , we see that the projection of is an ideal of , and is hence or as is simple. As is not contained in , we see that surjects onto ; similarly it surjects onto . An analogous argument shows that the intersection of with is either or ; the latter would force by the surjective projection onto , which was already ruled out. Thus has trivial intersection with , and similarly with , and is thus a graph. Such a graph cannot be an ideal of , so that . As was a subalgebra that surjected onto both and , we conclude by arguing as before that is also a graph; as is a Lie algebra, the graph is that of a Lie algebra isomorphism by the Lie algebra closed graph theorem (see this previous blog post). Since , we see that restricts to the graph of the identity on , and the claim follows.

Remark 7 The above arguments show that every root can be obtained from the maximal root by iteratively subtracting off simple roots (while staying in ), which among other things implies that the maximal root is unique. These facts can also be established directly from the axioms of a root system (or from the classification of root systems), but we will not do so here. By using Theorem 29, one can convert graph automorphisms of the Dynkin diagram (e.g. the automorphism sending the Dynkin diagram to its inverse, or the triality automorphism that rotates the diagram) to automorphisms of the Lie algebra; these are important in the theory of twisted groups of Lie type, and more specifically the Steinberg groups and Suzuki-Ree groups, but will not be discussed further here.

Remark 8 In a converse direction, once one establishes that in an irreducible root system that every root can be obtained from the maximal root by subtracting off simple roots (while staying in ), this shows that any Lie algebra associated to this system is necessarily simple. Indeed, given any non-trivial ideal in and a non-trivial element of , one locates a minimal element of in which has a non-trivial component, then iteratively applies raising operators to then locate a non-trivial element of the root space of the maximal root in ; if one then applies lowering operators one recovers all the other root spaces, so that .

Theorem 29, when combined with the results from previous sections, already gives Theorem 2, but without a fully explicit way to determine the Lie algebras listed in that theorem (or even to establish whether these systems exist at all). In the case of the classical Lie algebras , one can explicitly describe these algebras in terms of the special linear algebras , special orthogonal algebras , and symplectic algebras , but this does not give too much guidance as to how to explicitly describe the exceptional Lie algebras . We now turn to the question of how to explicitly describe all the simple Lie algebras in a unified fashion.

Let be a simple Lie algebra, with Cartan algebra . We view as a Hilbert space with the Killing form, and then identify this space with its dual . Thus for instance the coroot of a root is now given by the simpler formula

Let be the root system, which is irreducible. As described in Section 6, we have the vector space decomposition

where the spaces are one-dimensional, thus we can choose a generator for each , though we have the freedom to multiply each by a complex constant, which we will take advantage of to perform various normalisations. A basis for algebra together with the then form a basis for , known as a Cartan-Weyl basis for this Lie algebra. From (11), (20) we have

where is the quantity

which is always an integer because is a root system (indeed takes values in , and form an interesting matrix known as the Cartan matrix).

As discussed in Section 6, is a multiple of the coroot ; by adjusting for each pair we may normalise things so that

for all (here we use the fact that to avoid inconsistency). Next, we see from (19) that

if , and

for some complex number if . By considering the action of on (16) using Theorem 16 one can verify that is non-zero; however, its value is not yet fully determined because there is still residual freedom to normalise the . Indeed, one has the freedom to multiply by any non-zero complex scalar as long as (to preserve the normalisation (21)), in which case the structure constant gets transformed according to the law

However, observe that the combined structure constant is unchanged by this rescaling. And indeed there is an explicit formula for this quantity:

Lemma 30 For any roots with , one has

where are the string of roots of the form for integer .

This formula can be confirmed by an explicit computation using Theorem 16 (using, say, the standard basis for to select , which then fixes by (21)); we omit the details.

On the other hand, we have the following clever renormalisation trick of Chevalley, exploiting the abstract isomorphism from Theorem 29:

Lemma 31 (Chevalley normalisation) There exist choices of such that

for all roots with .

Proof: We first select arbitrarily, then we will have

for some non-zero for all roots . The plan is then to locate coefficients so that the transformation (23) eliminates all of the factors.

To do this, observe that we may identify with itself and with itself via the negation map for and for . From this and Theorem 29, we may find a Lie algebra isomorphism that maps to on , and thus maps to for any root . In particular, we have

for some non-zero coefficients ; from (21) we see in particular that

If we then apply to (22), we conclude that

when is a root, so that takes the special form

If we then select so that

for all roots (this is possible thanks to (24)), then the transformation (23) eliminates as desired.

From the above two lemmas, we see that we can select a special Cartan-Weyl basis, known as a Chevalley basis, such that

whenever is a root; in particular, the structure constants are all integers, which is a crucial fact when one wishes to construct Lie algebras and Chevalley groups over fields of arbitrary characteristic. This comes very close to fully describing the Lie algebra structure associated to a given Dynkin diagram, except that one still has to select the signs in (25) so that one actually gets a Lie algebra (i.e. that the Jacobi identity (1) is obeyed). This turns out to be non-trivial; see this paper of Tits for details. (There are other approaches to demonstrate existence of a Lie algebra associated to a given root system; one popular one proceeds using the Chevalley-Serre relations, see e.g. this text of Serre. There is still a certain amount of freedom to select the signs, but this ambiguity can be described precisely; see the book of Carter for details.) Among other things, this construction shows that every root system actually creates a Lie algebra (thus far we have only established uniqueness, not existence), though once one has the classification one could also build a Lie algebra explicitly for each Dynkin diagram by hand (in particular, one can build the simply laced classical Lie algebras and the maximal simply laced exceptional algebra , and construct the remaining Lie algebras by taking fixed points of suitable involutions; see e.g. these notes of Borcherds et al. for this approach).

An important precursor of the CFSG is the Feit-Thompson theorem from 1962-1963, which asserts that every finite group of odd order is solvable, or equivalently that every non-abelian finite simple group has even order. This is an immediate consequence of CFSG, and conversely the Feit-Thompson theorem is an essential starting point in the proof of the classification, since it allows one to reduce matters to groups of even order for which key additional tools (such as the Brauer-Fowler theorem) become available. The original proof of the Feit-Thompson theorem is 255 pages long, which is significantly shorter than the proof of the CFSG, but still far from short. While parts of the proof of the Feit-Thompson theorem have been simplified (and it has recently been converted, after six years of effort, into an argument that has been verified by the proof assistant Coq), the available proofs of this theorem are still extremely lengthy by any reasonable standard.

However, there is a significantly simpler special case of the Feit-Thompson theorem that was established previously by Suzuki in 1957, which was influential in the proof of the more general Feit-Thompson theorem (and thus indirectly to the proof of CFSG). Define a CA-group to be a group with the property that the centraliser of any non-identity element is abelian; equivalently, the commuting relation (defined as the relation that holds when commutes with , thus ) is an equivalence relation on the non-identity elements of . Trivially, every abelian group is CA. A non-abelian example of a CA-group is the group of invertible affine transformations on a field . A little less obviously, the special linear group over a finite field is a CA-group when is a power of two. The finite simple groups of Lie type are not, in general, CA-groups, but when the rank is bounded they tend to behave as if they were “almost CA”; the centraliser of a generic element in , for instance, when is bounded and is large), is typically a maximal torus (because most elements in are regular semisimple) which is certainly abelian. In view of the CFSG, we thus see that CA or nearly CA groups form an important subclass of the simple groups, and it is thus of interest to study them separately. To this end, we have

Theorem 1 (Suzuki’s theorem on CA-groups) Every finite CA-group of odd order is solvable.

Of course, this theorem is superceded by the more general Feit-Thompson theorem, but Suzuki’s proof is substantially shorter (the original proof is nine pages) and will be given in this post. (See this survey of Solomon for some discussion of the link between Suzuki’s argument and the Feit-Thompson argument.) Suzuki’s analysis can be pushed further to give an essentially complete classification of all the finite CA-groups (of either odd or even order), but we will not pursue these matters here.

Moving even further down the ladder of simple precursors of CSFG is the following theorem of Frobenius from 1901. Define a Frobenius group to be a finite group which has a subgroup (called the Frobenius complement) with the property that all the non-trivial conjugates of for , intersect only at the origin. For instance the group is also a Frobenius group (take to be the affine transformations that fix a specified point , e.g. the origin). This example suggests that there is some overlap between the notions of a Frobenius group and a CA group. Indeed, note that if is a CA-group and is a maximal abelian subgroup of , then any conjugate of that is not identical to will intersect only at the origin (because and each of its conjugates consist of equivalence classes under the commuting relation , together with the identity). So if a maximal abelian subgroup of a CA-group is its own normaliser (thus is equal to ), then the group is a Frobenius group.

Frobenius’ theorem places an unexpectedly strong amount of structure on a Frobenius group:

Theorem 2 (Frobenius’ theorem) Let be a Frobenius group with Frobenius complement . Then there exists a normal subgroup of (called the Frobenius kernel of ) such that is the semi-direct product of and .

Roughly speaking, this theorem indicates that all Frobenius groups “behave” like the example (which is a quintessential example of a semi-direct product).

Note that if every CA-group of odd order was either Frobenius or abelian, then Theorem 2 would imply Theorem 1 by an induction on the order of , since any subgroup of a CA-group is clearly again a CA-group. Indeed, the proof of Suzuki’s theorem does basically proceed by this route (Suzuki’s arguments do indeed imply that CA-groups of odd order are Frobenius or abelian, although we will not quite establish that fact here).

Frobenius’ theorem can be reformulated in the following concrete combinatorial form:

Theorem 3 (Frobenius’ theorem, equivalent version) Let be a group of permutations acting transitively on a finite set , with the property that any non-identity permutation in fixes at most one point in . Then the set of permutations in that fix no points in , together with the identity, is closed under composition.

Again, a good example to keep in mind for this theorem is when is the group of affine permutations on a field (i.e. the group for that field), and is the set of points on that field. In that case, the set of permutations in that do not fix any points are the non-trivial translations.

To deduce Theorem 3 from Theorem 2, one applies Theorem 2 to the stabiliser of a single point in . Conversely, to deduce Theorem 2 from Theorem 3, set to be the space of left-cosets of , with the obvious left -action; one easily verifies that this action is faithful, transitive, and each non-identity element of fixes at most one left-coset of (basically because it lies in at most one conjugate of ). If we let be the elements of that do not fix any point in , plus the identity, then by Theorem 3 is closed under composition; it is also clearly closed under inverse and conjugation, and is hence a normal subgroup of . From construction is the identity plus the complement of all the conjugates of , which are all disjoint except at the identity, so by counting elements we see that

As normalises and is disjoint from , we thus see that is all of , giving Theorem 2.

Despite the appealingly concrete and elementary form of Theorem 3, the only known proofs of that theorem (or equivalently, Theorem 2) in its full generality proceed via the machinery of group characters (which one can think of as a version of Fourier analysis for nonabelian groups). On the other hand, once one establishes the basic theory of these characters (reviewed below the fold), the proof of Frobenius’ theorem is very short, which gives quite a striking example of the power of character theory. The proof of Suzuki’s theorem also proceeds via character theory, and is basically a more involved version of the Frobenius argument; again, no character-free proof of Suzuki’s theorem is currently known. (The proofs of Feit-Thompson and CFSG also involve characters, but those proofs also contain many other arguments of much greater complexity than the character-based portions of the proof.)

It seems to me that the above four theorems (Frobenius, Suzuki, Feit-Thompson, and CFSG) provide a ladder of sorts (with exponentially increasing complexity at each step) to the full classification, and that any new approach to the classification might first begin by revisiting the earlier theorems on this ladder and finding new proofs of these results first (in particular, if one had a “robust” proof of Suzuki’s theorem that also gave non-trivial control on “almost CA-groups” – whatever that means – then this might lead to a new route to classifying the finite simple groups of Lie type and bounded rank). But even for the simplest two results on this ladder – Frobenius and Suzuki – it seems remarkably difficult to find any proof that is not essentially the character-based proof. (Even trying to replace character theory by its close cousin, representation theory, doesn’t seem to work unless one gives in to the temptation to take traces everywhere and put the characters back in; it seems that rather than abandon characters altogether, one needs to find some sort of “robust” generalisation of existing character-based methods.) In any case, I am recording here the standard character-based proofs of the theorems of Frobenius and Suzuki below the fold. There is nothing particularly novel here, but I wanted to collect all the relevant material in one place, largely for my own benefit.

— 1. Basic character theory —

Let be a finite group. Then we can form the finite-dimensional complex Hilbert space of functions with the inner product

and thus norm

where is the averaging operator. Inside this space, we have the subspace of class functions: functions which are invariant under the conjugation action of on itself, thus

for all . Equivalently, is constant on each conjugacy class of . In particular, we see that the dimension of is equal to the class number of – the number of conjugacy classes of .

One way to generate class functions is from taking traces of finite-dimensional unitary representations , i.e. a homomorphism from the group to unitary operators on a finite-dimensional complex Hilbert space . We will abbreviate “finite-dimensional unitary representation” as “representation” henceforth. Given any such representation, one has an associated character defined by

One easily verifies that this is a class function. For instance, the regular representation , in which , has character

and every linear character (i.e. a homomorphism to the complex unit circle) is a character associated the obvious one-dimensional representation corresponding to . In particular, the constant function is a character, associated to the principal one-dimensional representation.

The characters interact well with various representation-theoretic operations. For instance, isomorphic representations clearly have the same character. If are representations, then the characters of the direct sum and tensor product are the sum and product of the individual characters:

and

Also, if is the Hilbert space with the conjugated inner product, then the conjugate representation (given by taking and conjugating the inner product structure) has a conjugated character:

Thus the space of characters forms a semi-ring in that is closed under complex conjugation. Also, since any element of the absolute Galois group of the rationals can be extended to the complex numbers , we have the stronger fact that the space of characters is also invariant with respect to the action of ; we will need this fact somewhat later in this post.

The space of characters is not a ring, because characters are certainly not preserved with respect to negation: the value of a representation at the identity is the dimension of the space that acts on, and so is non-negative; in particular, will not be a character as long as is positive dimensional. We can then define the space of generalised characters to be the ring generated by the characters, thus a generalised character is nothing more than a difference of two characters.

By repeatedly taking orthogonal complements, one can easily see that representations are completely reducible, thus if is a collection of one representative of each of the isomorphism classes of irreducible finite-dimensional representations of , then every character can be written as a linear combination (over the natural numbers) of the irreducible characters .

From the ergodic theorem (which is a triviality in the case of an action of a finite group ), the average value of a character of a representation is equal to the dimension of its invariant component :

As a consequence, the inner product of two characters is equal to the dimension of the -invariant component of . By Schur’s lemma, this implies in particular that the irreducible characters are an orthonormal system in :

In fact, they form an orthonormal basis for . (Proof: given any non-trivial , the convolution operator is non-trivial in the regular representation, and thus must also be non-trivial with respect to at least one irreducible representation , which implies that has non-zero inner product with . Thus there is no non-zero element of that is orthogonal to all the irreducible characters, giving the claim.) In particular, we see that is equal to the class number of .

Using this basis, we now have a Fourier transform that is an isometry between the Hilbert space of class functions, and the Hilbert space , defined by taking inner products with characters

and with the usual Fourier inversion formula

and Plancherel and Parseval identities

and

(One can also relate convolution in with pointwise multiplication in , and pointwise multiplication in is related to a plethysm on involving tensor product multiplicities, but we will not need these operations here.)

A character of a (not necessarily irreducible) representation then has Fourier coefficients that count the multiplicity of in ; in particular, a class function is a character (resp. generalised character) iff its Fourier coefficients are all natural numbers (resp. integers), and any two representations are isomorphic iff they have the same character.

Thus for instance the regular representation has Fourier coefficients , leading to the identity

which gives the Peter-Weyl theorem that is isomorphic to the direct sum of copies of for each . In particular we have

From these Fourier identities we can now detect whether a representation is irreducible (or a combination of a small number of representations) through the structure of its character. Indeed, by taking Fourier transforms and working in we now have the following immediate corollaries, which will be very useful to us in the sequel:

Lemma 4 (Small norm and irreducibility) Let be a generalised character.

• (i) is a natural number.
• (ii) equals iff for some sign and irreducible character . If we also know that , this forces (since is positive).
• (iii) equals iff for some signs and distinct irreducible characters . If we also know that , then this forces (again because are positive).

One can of course also characterise when is equal to , , etc. by this method, although the descriptions rapidly become more complicated and less useful. In practice, this lemma will allow us to construct interesting examples of irreducible representations by first exhibiting a generalised character of small norm (or equivalently, two characters that are close to each other in norm). It seems very difficult to mimic this type of construction by any other means, including non-character-based representation theoretic methods. (But perhaps one could categorify this lemma somehow using K-theory.)

Lemma 4 is a typical application of the integrality gap, which is the trivial but fundamental fact that integers are either zero or have magnitude at least one. It is the interplay between the integrality gap and the Fourier analysis of characters which drives the proof of both Frobenius’ theorem and Suzuki’s theorem, as we shall soon see.

We also record some additional easy properties of characters which we will need later. Firstly, we have the identity

for any , since the inverse of a unitary operator is also its adjoint. Secondly, the kernel

of a character is automatically a normal subgroup of . This is because a unitary operator on a finite-dimensional space has trace if and only if it is equal to the identity operator, and so (3) is also the kernel of the associated representation . This latter fact suggests a strategy to prove Frobenius’ theorem by exhibiting a character whose kernel is precisely the complement of the conjugacy classes of excluding the identity, and this is in fact exactly what we will do.

Thus far we have focused on the representation theory of a single group . However, the situation becomes significantly more interesting when one relates the representation theory of two groups, a finite group and a subgroup (not necessarily normal). We then have an obvious restriction map that restricts any class function on to a class function on (since any two elements of that are conjugate in are clearly also conjugate in ). The adjoint map can be easily computed: given any class function , the induced class function is given by the Frobenius formula

with the convention that is extended by zero from to . Equivalently, one has

where is an enumeration of the left-cosets of .

In a similar fashion, given a representation of , we may restrict it to to obtain a representation of . In the adjoint direction, given a representation of , one can associate an induced representation of by the following construction. One takes to be the space of functions that for all and with inner product

and then lets act on by the formula

which one can check to indeed give a representation. Thus, for instance, the regular representation on induces (an isomorphic copy of) the regular representation on . It is not difficult to show that these representation-theoretic constructions are compatible with the operations on class functions mentioned earlier, thus

for every representation of , and dually

In particular, any character of restricts to a character of , and every character of induces a character on . Thus the adjoint relationship between and for class functions induces a corresponding adjoint relationship for representations, known as Frobenius reciprocity.

In general, the restriction or induction of an irreducible representation will not be irreducible (and the operations of restriction and induction do not invert each other). However, the geometry of the characters can often be controlled quite precisely (especially for structured groups such as Frobenius groups or CA-groups), and this together with tools such as Lemma 4 can allow us to create interesting irreducible representations of in non-trivial fashion from irreducible representations of .

— 2. Frobenius’ theorem —

We are now ready to prove Frobenius’ theorem. Let be a Frobenius group with Frobenius complement . Then there are distinct conjugates of , which are all disjoint except for the origin, thus one can partition

where (by abuse of notation) ranges over a set of representatives of the left cosets of and is the identity together with all the elements that do not lie in any conjugate of . I like to think of this decomposition by picturing as being like a plane, being the origin in this plane, each conjugate being a non-vertical line through the origin, and being the vertical line through the origin (note that in the case of the group, this is more or less exactly what (5) actually looks like). Counting elements in (5), we thus have

and so

We now start inducing characters from to and determine their geometr;8 Let be a irreducible character of of some dimension , then the character obeys the identities

and

In particular,

Now we consider the induced character

Using the above identities and the partition (5), we see that equals at , vanishes at all other elements of , and on each of the sets is given by a conjugate of . In particular,

Similarly, if one induces the trivial character on to a character on , this character will equal at , vanish at all other elements of , and will equal on . If we then form the generalised character

then equals on and is equal to outside of . In particular, we have

Using (6), we thus see that has surprisingly small norm:

We can then apply Lemma 4 and conclude that for some irreducible representation of . Note that . Thus we have shown that every irreducible representation of is the restriction of an irreducible representation of .

Remark 1 The above analysis shows a little bit more, namely that arises in as the orthogonal complement of a copy of the mean zero component of quasiregular representation on (i.e. the induction of the trivial representation on ), although it is not obvious to me how one would demonstrate (other than via an inspection of characters) that the induced representation actually contains a copy of this component of the quasiregular representation.

Now we consider the regular character of :

This character equals at the identity, and vanishes at the other elements of . If we then form the associated character

of , then restricts to and so also equals at the identity and vanishes at the other elements of . By (5), we conclude that (which is a class function) is supported on . Also, as each of the are constant on , is also, and so equals on all of . Thus is the kernel (3) of the character and is thus normal. This gives Frobenius’ theorem as discussed in the introduction.

— 3. More character theory —

Before we turn to Suzuki’s theorem, we will need some additional facts about characters which go beyond the Fourier-analytic considerations of Section 1 by also employing some tools from algebraic number theory.

Let be a finite group. Observe that if is a representation, then for any , the unitary operator can be diagonalised. As (and hence ) has finite order, the eigenvalues of are roots of unity, and so the trace is the sum of finitely many roots of unity. In particular, is always an algebraic integer. Unlike rational integers, algebraic integers do not directly enjoy an integrality gap; one can have algebraic integers of arbitrarily small nonzero magnitude (e.g. powers of ). However, we will rely in several places on the basic but fundamental fact that a number which is both an algebraic integer and a rational is necessarily a rational integer, which then is subject to the integrality gap.

We have a variant of the above fact:

Lemma 5 Let be a finite group, let be a -dimensional irreducible representation, and let . Then is an algebraic integer, where is the conjugacy class of .

Proof: The endomorphism is -equivariant and has trace ; by Schur’s lemma, it is thus equal to times the identity. It thus suffices to show that the diagonal entries of are algebraic integers; thus it will suffice to show that for some monic polynomial with integer coefficients.

Consider the associated element in the group ring of . Then the modules , , , etc. form an increasing sequence of submodules of . As is Noetherian, we thus have

for some , or equivalently that is an integer combination of . This implies that is an integer combination of , and the claim follows.

This leads to an important corollary:

Corollary 6 (Dimension divides order) Let be a finite group, and let be an irreducible -dimensional representation. Then divides .

Proof: As the character has norm one, we have

Grouping the summation by conjugacy classes , we can express the left-hand side as the sum of terms of the form , which by the preceding lemma and discussion is equal to times an algebraic integer. We conclude that is an algebraic integer also; but it is rational, and so must be a rational integer also.

In particular, if is of odd order, then the dimension of any irreducible representation of has odd dimension. This has a further important consequence, due to Burnside:

Proposition 7 (Odd groups have non-real characters) Let be a finite group of odd order, and let be a non-principal irreducible character of . Then is not a real-valued character. In other words, .

Proof: Suppose for contradiction that is real-valued. By (2) this implies that for all .

As is odd, there are no elements in of order . Thus one can partition

for some subset of of order . On the other hand, as is non-principal, we have

By (7) one has

But is the dimension of , which is odd by Corollary 6, so is a half-integer. But it is also an algebraic integer, giving the desired contradiction.

— 4. Suzuki’s theorem —

We can now begin the proof of Suzuki’s theorem; we will basically use an arrangement of this theorem from the thesis of Wilcox. We begin with an easy reduction to the simple case:

Proposition 8 (Reduction to the simple case) Let be a finite CA-group of odd order which is not simple. Suppose that all CA groups of smaller odd order than are solvable. Then is solvable also.

Proof: If is not simple, it has a proper normal subgroup . This group is also of odd order and inherits the CA property from , so by hypothesis is solvable. If we let be the last non-trivial group in the derived series of , then is a non-trivial abelian characteristic subgroup of , and is thus also a normal subgroup of . Let be the centraliser of , then is also a normal subgroup of , which is still non-trivial and abelian as is a CA-group. Furthermore, is maximal abelian (it is not contained in any larger abelian group).

To show that is solvable, it then suffices to show that the quotient is solvable. As this group has an odd order smaller than that of , it suffices to show that is a CA-group. Thus, if are non-identity elements of with both commuting with , we need to show that commute with each other. Equivalently, if are such that both commute with modulo , then commutes with modulo .

If we fix , then acts on by conjugation. This action cannot fix any non-identity element of , else the centraliser of would contain as well as , contradicting the maximal abelian nature of . Thus the map , which is a homomorphism on the normal abelian group , has trivial kernel and is thus an isomorphism. From this we see that if commutes with modulo , then one can multiply by an element of (on the left or right) in order to make it commute with exactly. Thus, without loss of generality, and both commute with exactly, and so commute exactly as well as is a CA-group, giving the claim.

In view of this proposition, we see that to prove Suzuki’s theorem it suffices to show that simple non-abelian CA-groups of odd order do not exist.

Observe that in a CA-group , every non-identity element of is contained in a unique maximal abelian subgroup of , namely the centraliser of . Thus the maximal abelian subgroups of , once one removes the identity, form a partition of . It is instructive to keep some examples in mind:

• In the case of the group on a field , the maximal abelian subgroups are the translation group and the stabilisers of points , where is the multiplicative group .
• In the case of the special linear group with a power of two, the maximal abelian groups are conjugates of the split torus

  

  the non-split torus

  

  (where is any quantity not of the form for some ) or the unipotent group

  

As these examples show, while many of the maximal abelian subgroups may be conjugate to each other, there can certainly be several non-conjugate examples of maximal abelian subgroups. Let be a set consisting of one representative from each of these conjugacy classes, then we have the following analogue of the partition (5):

where is the normaliser of . This partition turns out to be a somewhat less favourable than (5), but one can still run analogues of the Frobenius argument, particularly in the case when is odd (which forces many other related quantities, such as , to be odd also). I like to think of this decomposition by viewing as a plane, as the origin, and as sweeping out various sectors of this plane, with each conjugate of being one of the rays in the sector. (This picture is an oversimplification, for instance it does not accurately reflect the closure of with respect to group inversion, but I still find it a useful picture to have in mind.)

We first give the analogue of (6). Taking cardinalities in (8) we obtain the class equation

for a CA-group, which we can rearrange as

where is the group (I like to think of this group as a sort of “Weyl group” associated to ). As a first approximation, the right hand side of (9) is close to . Thus, if one can somehow prevent from getting too small for too many values of , one an hope to upper bound the right-hand side of (9) by something less than , leading to the desired contradiction. (This strategy won’t quite work when is very small – to be precise – but this case can be worked out by hand.) Thus, we will be looking for such things as lower bounds on or upper bounds on . The fact that is odd will force the and to be odd as well, which will turn out to be useful in improving these bounds by doubling the power of the integrality gap. (We will also rely on the odd order of in a number of other places, in particular using Proposition 7.)

To get started on this strategy, suppose first that was trivial for some , thus , and then all the conjugates of are distinct. This makes a Frobenius group, and so by Theorem 2 there is a Frobenius kernel , which is a normal subgroup of . If is simple, then has to be trivial, which makes and so is abelian. This gives Suzuki’s theorem in this case. Thus we may assume that for all ; as the are all odd, we may improve this to

for all . From this and (9) (and bounding by one of the ) we thus see that is not too small:

Next, we use the Sylow theorems to make the pairwise coprime:

Lemma 9 For any distinct , and are coprime.

Proof: Suppose for contradiction that and are divisible by a common prime . Then and both contain groups of order , and thus both non-trivially intersect a Sylow -group. On the other hand, non-trivial Sylow -groups have non-trivial centre (otherwise all conjugacy classes other than the identity would have order divisible by , contradiction) and so must be abelian in a CA group. By further application of the CA property we thus conclude that and both contain a Sylow -group. But all Sylow -groups are conjugate, and so non-trivially intersects a conjugate of , contradicting (8).

We remark that the above analysis also reveals that

(because the order of a Sylow -group is the largest power of dividing ), although we will not need to rely on this fact here. (Actually we will barely use Lemma 9 as it is, it being needed to dispose of one technical case in the final analysis.) It is interesting though to see that classical techniques such as Sylow theorems are capable of demonstrating a number of facts about the various quantities appearing in the class equation (9), although without the additional control arising from character theory these facts appear to be insufficient in and of themselves to actually contradict that equation. As an example of (12) one can take the special linear group with a power of two, in which there are three abelian groups (split torus, non-split torus, and unipotent group) of orders respectively, with the entire group being of order .

We do not yet have a sufficiently strong upper bound on the right-hand side of (9), basically because we have no upper bound on the number of conjugacy classes (or sufficiently good lower bounds on the ). To get further bounds we have to return to character theory. The basic idea will be to construct generalised characters which have small norm but which take non-trivial rational integer values at many places, which when combined with the integrality gap will yield useful bounds on various quantities that appear in (9).

We turn to the details. Let be one of the maximal abelian groups. Being abelian, the character theory of is just Fourier analysis: is the group of linear characters on (i.e. the Pontryagin dual of ). (Indeed, from (1) and the observation that the class number of an abelian group is the same as its order, we see that all irreducible representations of an abelian group are one-dimensional.)

The normaliser acts on by conjugation; as is abelian, the action of on itself is trivial, and so we obtain an action of on also. Taking adjoints, we obtain an action of on as well. Any non-trivial element of cannot fix an non-trivial element of , as the centraliser of would then contain an element outside of , contradicting the CA-group nature of . Taking adjoints, we conclude that a non-trivial element of cannot fix an non-trivial element of either (otherwise the action of minus the identity would be non-injective, hence non-surjective, on , so that the corresponding homomorphism on is non-injective). Thus we see that the action of foliates the non-identity elements of into orbits of size . Among other things, this implies that divides , and that the number of such orbits is . As are both odd, is even, and in particular

for all .

Now let us make some generalised characters. Let be non-identity elements of which do not lie in the same -orbit. Then is a generalised character of that vanishes at the identity. Applying induction and (4), we see that

is a generalised character of that is supported on the set

and whose restriction to takes the form

From the Plancherel identity (and the assumption that have disjoint -orbits) we see that

and so

since , we thus have

We can then apply Lemma 4 and conclude the important fact that is the difference of two distinct irreducible characters of .

Let be a set of representatives of all the orbits of . Then by the above discussion, we see that

is the difference of two distinct irreducible characters of whenever are distinct. From the linear independence of the irreducible characters and some easy combinatorics (using (13)), we then see that we can find distinct irreducible characters of and a sign such that

for all . For , the sign and the characters are unique. When , there is a non-uniqueness: one then has the freedom to swap while reversing the sign of . But the set of characters remains unique. We will call the the exceptional characters associated to .

Note that if is in the absolute Galois group of the rationals, then permutes the non-trivial linear characters of . Applying to (15) and using the uniqueness of the set of exceptional characters, we conclude that also permutes the exceptional characters of . On the other hand, from (15) we know that the exceptional characters of all agree outside of , and are thus fixed by the absolute Galois group in this region; in other words, they are rational outside of . On the other hand, as mentioned in the previous section, characters always take the values of algebraic integers. We conclude that

Lemma 10 Any exceptional character for takes rational integer values outside of .

As remarked earlier, from (15) we see that is supported on the set . In particular, if and are distinct, and and , then and are orthogonal. From this and the orthonormality of irreducible characters, we conclude (again using (13)) that and are distinct. Thus we see that the total number of exceptional characters in ; together with the trivial character, this gives distinct irreducible characters of . On the other hand, observe that , and hence , consists of conjugacy classes, and so from (8) we see that the class number of is also . As these numbers match, we see that we have located all of the irreducible characters of ; thus every non-principal irreducible character of is an exceptional character for some .

Now that we have identified the irreducible characters of , we can analyse other generalised characters in terms of them. We pick and consider the generalised character

As with (14), is supported on and when restricted to , is equal to

In particular, from Plancherel’s theorem we have

and hence (since )

This is a bit too large of a norm to apply Lemma 4 again, but is still only of moderate size (recall our enemy when trying to contradict (9) is that the are too small, too frequently), and we can nevertheless use the geometry of and the other known characters, together with the integrality gap, to limit how breaks up into irreducible components. Firstly, since the all have mean zero, we see that (16) sums to on , and thus

Thus the Fourier coefficient of at the trivial representation is . Next, we see that (16) is orthogonal to for any and , which upon summing on the conjugacy classes of and on gives that

Thus the Fourier coefficients of at the exceptional characters for are all equal. Similarly, we have

for any and , so from (15) we have

and so the Fourier coefficient at is plus the Fourier coefficients at all the other exceptional characters at . Next, for distinct from and , we see from (15) that the generalised character is supported on , which by (8) is disjoint from the support of , thus

Thus all the Fourier coefficients of at exceptional characters of are the same. We thus have obtained a decomposition of the form

for some natural numbers .

Taking norms using the orthonormality of the irreducible characters, we conclude that

Note that regardless of what is, the quantity is always at least one, thanks to (13). We thus obtain an upper bound on the :

In particular, from (13) we see that there are not many for which is non-zero:

This is progress towards our goal of bounding (9) (because it helps control ), except that we also need to deal with those for which is zero. For this, the generalised character will no longer be useful, but another character of small norm – namely, – will be available as a substitute.

We turn to the details. Let be such that . Then we return to (18) and conclude that

on the set . On this set, we know from Lemma 10 that is an integer. But furthermore, from Proposition 7 we know that the exceptional characters come in conjugate pairs, so in fact Lemma 10 gives that is an even integer. We conclude that is an odd integer on , and in particular, has magnitude at least on this set. As all the agree outside of , we conclude that

on . On the other hand, from the orthonormality of the we know that has an norm of . Since each set has cardinality (as was shown in the derivation of the class equation (9)), we conclude that

We now have enough bounds on the various terms in (9) to obtain the necessary contradiction to finish Suzuki’s theorem from an elementary (though admittedly ad hoc) analysis. It is convenient to order the subgroups so that

We then write the right-hand side of (9) as

For the first summation, we crudely bound by and use (19); for the second summation we use (20). We conclude from (9) that

and thus (bounding by and by )

Applying (10), we conclude that , which forces since is even. Since , we conclude that . We use this to return to the bound (22) to obtain

and hence

If , then

and so

and so

On the other hand

The two bounds are inconsistent for , so we have from (10) (and the odd nature of ), which then gives the upper bound from (23), and Suzuki’s theorem can be verified by classical computations for the odd non-abelian groups of order less than (of which there are actually not that many); alternatively one can use (11), Lemma 9, and (12) to eliminate this case (as no odd number less than has more than three prime factors). So the only remaining case is when and . In this case we may interchange the indices and (which does not affect (21)) and repeating the above arguments we may thus also assume that . Since , we conclude that . But this contradicts Lemma 9, and Suzuki’s theorem is proved.

On the other hand, the basic object of study in dynamics (and in related fields, such as ergodic theory) is that of a dynamical system , that is to say a space together with a shift map (which is often assumed to be invertible, although one can certainly study non-invertible dynamical systems as well). One often adds additional structure to this dynamical system, such as topological structure (giving rise topological dynamics), measure-theoretic structure (giving rise to ergodic theory), complex structure (giving rise to complex dynamics), and so forth. A dynamical system gives rise to an action of the natural numbers on the space by using the iterates of for ; if is invertible, we can extend this action to an action of the integers on the same space. One can certainly also consider dynamical systems whose underlying group (or semi-group) is something other than or (e.g. one can consider continuous dynamical systems in which the evolution group is ), but we will restrict attention to the classical situation of or actions here.

There is a fundamental correspondence principle connecting the study of strings (or subsets of natural numbers or integers) with the study of dynamical systems. In one direction, given a dynamical system , an observable taking values in some alphabet , and some initial datum , we can first form the forward orbit of , and then observe this orbit using to obtain an infinite string . If the shift in this system is invertible, one can extend this infinite string into a doubly infinite string . Thus we see that every quadruplet consisting of a dynamical system , an observable , and an initial datum creates an infinite string.

Example 1 If is the three-element set with the shift map , is the observable that takes the value at the residue class and zero at the other two classes, and one starts with the initial datum , then the observed string becomes the indicator of the multiples of three.

In the converse direction, every infinite string in some alphabet arises (in a decidedly non-unique fashion) from a quadruple in the above fashion. This can be easily seen by the following “universal” construction: take to be the set of infinite strings in the alphabet , let be the shift map

let be the observable

and let be the initial point

Then one easily sees that the observed string is nothing more than the original string . Note also that this construction can easily be adapted to doubly infinite strings by using instead of , at which point the shift map now becomes invertible. An important variant of this construction also attaches an invariant probability measure to that is associated to the limiting density of various sets associated to the string , and leads to the Furstenberg correspondence principle, discussed for instance in these previous blog posts. Such principles allow one to rigorously pass back and forth between the combinatorics of strings and the dynamics of systems; for instance, Furstenberg famously used his correspondence principle to demonstrate the equivalence of Szemerédi’s theorem on arithmetic progressions with what is now known as the Furstenberg multiple recurrence theorem in ergodic theory.

In the case when the alphabet is the binary alphabet , and (for technical reasons related to the infamous non-injectivity of the decimal representation system) the string does not end with an infinite string of s, then one can reformulate the above universal construction by taking to be the interval , to be the doubling map , to be the observable that takes the value on and on (that is, is the first binary digit of ), and is the real number (that is, in binary).

The above universal construction is very easy to describe, and is well suited for “generic” strings that have no further obvious structure to them, but it often leads to dynamical systems that are much larger and more complicated than is actually needed to produce the desired string , and also often obscures some of the key dynamical features associated to that sequence. For instance, to generate the indicator of the multiples of three that were mentioned previously, the above universal construction requires an uncountable space and a dynamics which does not obviously reflect the key features of the sequence such as its periodicity. (Using the unit interval model, the dynamics arise from the orbit of under the doubling map, which is a rather artificial way to describe the indicator function of the multiples of three.)

A related aesthetic objection to the universal construction is that of the four components of the quadruplet used to generate the sequence , three of the components are completely universal (in that they do not depend at all on the sequence ), leaving only the initial datum to carry all the distinctive features of the original sequence. While there is nothing wrong with this mathematically, from a conceptual point of view it would make sense to make all four components of the quadruplet to be adapted to the sequence, in order to take advantage of the accumulated intuition about various special dynamical systems (and special observables), not just special initial data.

One step in this direction can be made by restricting to the orbit of the initial datum (actually for technical reasons it is better to restrict to the topological closure of this orbit, in order to keep compact). For instance, starting with the sequence , the orbit now consists of just three points , , , bringing the system more in line with the example in Example 1. Technically, this is the “optimal” representation of the sequence by a quadruplet , because any other such representation is a factor of this representation (in the sense that there is a unique map with , , and ). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system are interpreted as infinite strings rather than elements of a more geometrically or algebraically rich object (e.g. points in a circle, torus, or other homogeneous space).

For general sequences , locating relevant geometric or algebraic structure in a dynamical system generating that sequence is an important but very difficult task (see e.g. this paper of Host and Kra, which is more or less devoted to precisely this task in the context of working out what component of a dynamical system controls the multiple recurrence behaviour of that system). However, for specific examples of sequences , one can use an informal procedure of educated guesswork in order to produce a more natural-looking quadruple that generates that sequence. This is not a particularly difficult or deep operation, but I found it very helpful in internalising the intuition behind the correspondence principle. Being non-rigorous, this procedure does not seem to be emphasised in most presentations of the correspondence principle, so I thought I would describe it here.

The basic way to proceed here is to ask oneself the following information-theoretic question. Suppose you are given the sequence with values in some compact alphabet , and that some adversary selects a natural number without telling you exactly what it is. On the other hand, the adversary must truthfully answer any question about that you pose to it, provided that the answer takes values in some compact space (so one can for instance ask yes-no questions about , but one cannot simply ask the adversary what is, since the natural numbers are not compact). Your job is to acquire enough information from the adversary in order for you to determine not only the current value of the sequence, but also all subsequent values of the sequence. Furthermore, the answers that the adversary gives for a given choice of should be sufficient information to also determine what answers the adversary would also have given for . The space in which your answers reside in will eventually be the space in the quadruplet , with corresponding to the answers in that space that the adversary gives for a given choice of ; the observable is the function that takes the answers given and returns the value of that one can deduce from those answers, and the shift map encodes the relationship between the answers given for and the answers given for . (One can then often place an invariant measure on by considering (or guessing) the equidistribution properties of the orbit , but we will not pursue this point here.)

As it stands, this problem has a trivial answer: you simply asks the adversary for the entire tuple , which takes values in the compact space ; the adversary is forced to supply this information, which gives you everything you need for the problem, namely the value of and all subsequent elements of the sequence, as well as what the adversary would have provided for instead of . This corresponds to the universal construction outlined earlier. In the case of a binary alphabet , one can similarly ask for the real number in , which achieves the same goal (at least when the sequence does not end in an infinite string of s). (One could also ask the adversary for as embedded in a compactification of the natural numbers, such as the Stone-Cech compactification , to get another trivial solution to the problem; this gives rise to some other important ways to extract a dynamical interpretation to a set of integers. See this blog post for some variants on this theme.)

But the point of phrasing things this way is that one can look for more “elegant” ways in which to extract information from the adversary, in which one asks the adversary for a certain minimal amount of “relevant” information rather than asking what is basically an infinite number of questions rolled into one (which one then considers to be “cheating”). Ideally, the information asked should also reflect the underlying geometric or algebraic structure of the original sequence . From a purely rigorous mathematical viewpoint, there are no compelling objective criteria by which to judge what set of questions to ask the adversary is the “best” one, but one can nevertheless still proceed by relying on more informal and subjective evaluations of the information asked. It turns out that proceeding in an intelligently guided trial-and-error fashion, in which one starts with an initial first guess to the problem and refines it in reaction to the defects of that guess, is usually quite instructive in revealing the “true” dynamics behind any given sequence.

Let’s illustrate this with a few examples. We begin with the previously considered example of the indicator function of the multiples of three, so that equals when is a multiple of three and equals zero otherwise. Thus, it is natural as a first guess to simply ask the adversary if is a multiple of three. But while this question reveals to you the value of , it does not reveal the value of , because being told that is not a multiple of three is insufficient information to determine whether is a multiple of three. To put it another way, the information the adversary provided for is not sufficient to determine the information the adversary would have provided for . To resolve this, we soon see that we need a bit more information than whether is a multiple of three: we need the residue class of modulo three, and this is enough information both to work out , and to work out what the adversary would say for (since the residue class is simply the increment of the residue class ), which upon iteration then provides the value of , , etc. This gives the dynamical system described in Example 1.

Now let us consider another example. Let be a real number that is already known to you (e.g. one can take , for sake of concreteness), and consider the sequence , taking values in the interval . As before, the first naive guess here would be to simply ask the adversary for the value of , which of course will tell you what is, but does not quite reveal the value of . Indeed, the sine addition law gives

which does not quite allow us to determine from , because while and are known quantities, the cosine is not quite determined: the classic identity only determines up to a sign. This can be rectified in a number of ways. One is to ask the adversary for both and , as one can then update both of these pieces of data for through the addition law (1) as well as its counterpart

Using the usual identification of the unit circle with , one can reformulate this in a more modern fashion by simply asking the adversary for the value of in the unit circle , as this determines and can be updated using the shift map on . This interprets the sequence in terms of the circle shift dynamical system, in which and ; the observable in this case is and the initial datum is .

Now consider the sequence taking values now in , where are two given real numbers (e.g. and ). Pursuing a similar line of reasoning to the above, we soon see that a good set of questions to ask the adversary are the values of and , as these two pieces of information both determine the current member of the sequence, and can also be updated easily as one moves from to . This interprets the given sequence in terms of the -torus shift in which , , , and . More generally, any quasiperiodic sequence can be interpreted in terms of a shift map on a -torus.

Next, we consider the quadratic phase sequence . Based on previous experience, we can query the adversary for , but this is not sufficient by itself, because we cannot update this information into without additional input. However, since

we can attempt to fix this by also querying the adversary about (one could also use here if desired). Note that can itself be updated easily through the identity

By using the combination of these two queries, we represent the above quadratic phase sequence in terms of the skew shift system in which , , , and . Similarly for polynomial phase sequences, in which one replaces by some other polynomial of with real coefficients.

Now we turn to the example , where are real numbers and is the greatest integer less than or equal to . As before, we begin by querying the adversary for , but are then faced with the question of how to update this information when moving from to , that is to say we need a way to determine

in terms of queried data. The problem here is that could either equal or . But one can determine which of these is the case by querying for the fractional part of , or equivalently by querying for . Indeed, we see that

if , and

otherwise. This leads to the dynamical system whose shift map is given by setting

when , and

otherwise, with the observable and initial datum . This is not the only way to represent this sequence by a dynamical system, of course; if one observes that

then one can alternately proceed by querying for and , leading to a -torus system with , , , and . One can verify that this system is in fact conjugate to the previous system; it simplifies the shift at the expense of making the observable more complicated (in particular, at least one of the dynamics or the observable will be discontinuous).

Similar considerations apply to the example . Starting with a query to , one soon realises that to update this piece of information, one also needs access to and to . This gives rise to a mildly complicated system on a -torus (or, alternatively, a cube ) that tracks the fractional parts of , , and , with a piecewise polynomial shift function , the precise description of which we leave as an exercise to the reader. In analogy with the previous example, this system can be conjugated to a rotation on the Heisenberg nilmanifold

at the cost of turning the observable into a slightly messy piecewise smooth function; see e.g. this blog post for details. More generally, any function arising from phases that are bracket polynomials (or generalised polynomials) in (which means that, like , they are formed out of and some real constants using the ring operations and the floor function finitely many times), the associated sequences can be described in terms of a piecewise smooth function on a nilmanifold; see this paper of Bergelson and Leibman for details.

Thus far, we have restricted attention to sequences that are strongly “deterministic” in various senses (for instance, they have zero topological entropy, which roughly speaking means that one needs only queries in order to predict the behaviour of the sequence for the next steps in time). This is reflected in the nilpotent (and finite-dimensional) nature of the systems being produced. When the sequence is more chaotic, then it becomes difficult to find a dynamical system to describe the dynamics that is any simpler than the universal system. Consider for instance the sequence arising from the digits of in base , so that the alphabet here is . The problem here is that knowledge of the digit of tells us virtually nothing useful about the digit, and there is no obvious set of queries (other than that of asking for basically the entire string of digits after tne digit) that would let one perform an update from to . Thus, one is basically left with universal constructions, e.g. taking to be the interval , letting be the shift map (i.e. is the remainder of when divided by ), with observable and initial datum . It is widely expected that is a “generic” point of this system, which is equivalent to the notoriously difficult conjecture that is a normal number in base (of course, is conjectured to be normal in every base, but currently it is not known if it is normal in even one base). This dynamical system model does not shed too much light on this issue, since it is not tailored to in any way. One can try to use various explicit identities involving to hope to get a more useful dynamical system to model , but so far these efforts have not yielded dramatic results. (One could hope for instance that the Bailey-Borwein-Plouffe formula could help describe the base digits of , but the resulting dynamical system obtained is quite complicated.)

Now we turn to the sequence coming from the Möbius function (where we adopt for sake of discussion the convention that ). The Möbius function is conjectured to behave quite random in many ways (see this blog post for a discussion of some of these), and so one might think that there is no interesting dynamical system to model this function other than the universal ones. But one can perform a separation

of the Möbius function into the unsigned part of the Möbius function (which is when is square-free, and otherwise), and the Liouville function , which takes values in (assigning , say, for sake of discussion). Like the digits of , the Liouville function is widely expected to fluctuate in a chaotic fashion and is unlikely to be easily modeled by anything other than a universal model (e.g. the shift map on ). But the unsigned part is significantly more regular. Indeed, much as the indicator function of the multiples of three can be understood through the querying of , can be obtained after querying for all primes , since is precisely when all of these residue classes are non-zero. As each of these residue classes are easily updated when we increment to , we see that can be modeled by the dynamical system with with being the shift

and being the observable that sets to equal when the residue classes are all non-zero and otherwise, with initial datum . While this system does have positive topological entropy, it is otherwise fairly regular and can be analysed relatively easily (especially when compared with the Möbius or Liouville functions). For instance, an obvious invariant measure to place on this system is the Haar measure on , that is to say the product of the uniform probability measures on each of the . Using this measure, one can show using Fourier analysis that the shift map is uniquely ergodic, and that is an indicator function of a compact set of measure

Among other things, this shows that the square-free numbers have asymptotic density . Several further properties of this system were analysed recently by Cellarosi and Sinai. The Möbius function itself does not arise directly from this system, but instead comes from a joining of this system with a system modeling the Liouville function. Unfortunately, the latter system is poorly understood (one can use a universal model here, and conjecture that the Liouville function corresponds to a generic point in that model, but only a small number of aspects of this conjecture can be rigorous established with current technology), and our understanding of either the Möbius or Liouville functions are still very far from satisfactory.

There are a vast number of systems that lie intermediate between the deterministic systems associated to nilmanifolds, and the (conjecturally) chaotic systems associated to sequences such as the Möbius function or the digits of . For instance, the systems associated to automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence exhibit a number of interesting intermediate behaviours (e.g. the system associated to Thue-Morse has purely singular spectrum despite having zero topological entropy), and have attracted a substantial amount of attention in dynamics.

Proposition 1 (Additive rectification) Let be a subset of the additive group for some prime , and let be an integer. Suppose that . Then there exists a map into a subset of the integers which is a Freiman isomorphism of order in the sense that for any , one has

if and only if

Furthermore is a right-inverse of the obvious projection homomorphism from to .

The original version of the rectification principle allowed the sets involved to be substantially larger in size (cardinality up to a small constant multiple of ), but with the additional hypothesis of bounded doubling involved; see the above-mentioned papers, as well as this later paper of Green and Ruzsa, for further discussion.

The proof of Proposition 1 is quite short (see Theorem 3.1 of Bilu-Lev-Ruzsa); the main idea is to use Minkowski’s theorem to find a non-trivial dilate of that is contained in a small neighbourhood of the origin in , at which point the rectification map can be constructed by hand.

Very recently, Codrut Grosu obtained an arithmetic analogue of the above theorem, in which the rectification map preserves both additive and multiplicative structure:

Theorem 2 (Arithmetic rectification) Let be a subset of the finite field for some prime , and let be an integer. Suppose that . Then there exists a map into a subset of the complex numbers which is a Freiman field isomorphism of order in the sense that for any and any polynomial of degree at most and integer coefficients of magnitude summing to at most , one has

if and only if

Note that it is necessary to use an algebraically closed field such as for this theorem, in contrast to the integers used in Proposition 1, as can contain objects such as square roots of which can only map to in the complex numbers (once is at least ).

Using Theorem 2, one can transfer results in arithmetic combinatorics (e.g. sum-product or Szemerédi-Trotter type theorems) regarding finite subsets of to analogous results regarding sufficiently small subsets of ; see the paper of Grosu for several examples of this. This should be compared with the paper of Vu, Wood, and Wood, which introduces a converse principle that embeds finite subsets of (or more generally, a characteristic zero integral domain) in a Freiman field-isomorphic fashion into finite subsets of for arbitrarily large primes , allowing one to transfer arithmetic combinatorical facts from the latter setting to the former.

Grosu’s argument uses some quantitative elimination theory, and in particular a quantitative variant of a lemma of Chang that was discussed previously on this blog. In that previous blog post, it was observed that (an ineffective version of) Chang’s theorem could be obtained using only qualitative algebraic geometry (as opposed to quantitative algebraic geometry tools such as elimination theory results with explicit bounds) by means of nonstandard analysis (or, in what amounts to essentially the same thing in this context, the use of ultraproducts). One can then ask whether one can similarly establish an ineffective version of Grosu’s result by nonstandard means. The purpose of this post is to record that this can indeed be done without much difficulty, though the result obtained, being ineffective, is somewhat weaker than that in Theorem 2. More precisely, we obtain

Theorem 3 (Ineffective arithmetic rectification) Let . Then if is a field of characteristic at least for some depending on , and is a subset of of cardinality , then there exists a map into a subset of the complex numbers which is a Freiman field isomorphism of order .

Our arguments will not provide any effective bound on the quantity (though one could in principle eventually extract such a bound by deconstructing the proof of Proposition 4 below), making this result weaker than Theorem 2 (save for the minor generalisation that it can handle fields of prime power order as well as fields of prime order as long as the characteristic remains large).

Following the principle that ultraproducts can be used as a bridge to connect quantitative and qualitative results (as discussed in these previous blog posts), we will deduce Theorem 3 from the following (well-known) qualitative version:

Proposition 4 (Baby Lefschetz principle) Let be a field of characteristic zero that is finitely generated over the rationals. Then there is an isomorphism from to a subfield of .

This principle (first laid out in an appendix of Lefschetz’s book), among other things, often allows one to use the methods of complex analysis (e.g. Riemann surface theory) to study many other fields of characteristic zero. There are many variants and extensions of this principle; see for instance this MathOverflow post for some discussion of these. I used this baby version of the Lefschetz principle recently in a paper on expanding polynomial maps.

Proof: We give two proofs of this fact, one using transcendence bases and the other using Hilbert’s nullstellensatz.

We begin with the former proof. As is finitely generated over , it has finite transcendence degree, thus one can find algebraically independent elements of over such that is a finite extension of , and in particular by the primitive element theorem is generated by and an element which is algebraic over . (Here we use the fact that characteristic zero fields are separable.) If we then define by first mapping to generic (and thus algebraically independent) complex numbers , and then setting to be a complex root of of the minimal polynomial for over after replacing each with the complex number , we obtain a field isomorphism with the required properties.

Now we give the latter proof. Let be elements of that generate that field over , but which are not necessarily algebraically independent. Our task is then equivalent to that of finding complex numbers with the property that, for any polynomial with rational coefficients, one has

if and only if

Let be the collection of all polynomials with rational coefficients with , and be the collection of all polynomials with rational coefficients with . The set

is the intersection of countably many algebraic sets and is thus also an algebraic set (by the Hilbert basis theorem or the Noetherian property of algebraic sets). If the desired claim failed, then could be covered by the algebraic sets for . By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over cannot be covered by countably many varieties of smaller dimension, we conclude that must in fact be covered by a finite number of such sets, thus

for some . By the nullstellensatz, we thus have an identity of the form

for some natural numbers , polynomials , and polynomials with coefficients in . In particular, this identity also holds in the algebraic closure of . Evaluating this identity at we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows.

From Proposition 4 one can now deduce Theorem 3 by a routine ultraproduct argument (the same one used in these previous blog posts). Suppose for contradiction that Theorem 3 fails. Then there exists natural numbers , a sequence of finite fields of characteristic at least , and subsets of of cardinality such that for each , there does not exist a Freiman field isomorphism of order from to the complex numbers. Now we select a non-principal ultrafilter , and construct the ultraproduct of the finite fields . This is again a field (and is a basic example of what is known as a pseudo-finite field); because the characteristic of goes to infinity as , it is easy to see (using Los’s theorem) that has characteristic zero and can thus be viewed as an extension of the rationals .

Now let be the ultralimit of the , so that is the ultraproduct of the , then is a subset of of cardinality . In particular, if is the field generated by and , then is a finitely generated extension of the rationals and thus, by Proposition 4 there is an isomorphism from to a subfield of the complex numbers. In particular, are complex numbers, and for any polynomial with integer coefficients, one has

if and only if

By Los’s theorem, we then conclude that for all sufficiently close to , one has for all polynomials of degree at most and whose coefficients are integers whose magnitude sums up to , one has

if and only if

But this gives a Freiman field isomorphism of order between and , contradicting the construction of , and Theorem 3 follows.

Theorem 1 (Furstenberg-Sarkozy theorem) Let , and suppose that is sufficiently large depending on . Then every subset of of density at least contains a pair for some natural numbers with .

This theorem is of course similar in spirit to results such as Roth’s theorem or Szemerédi’s theorem, in which the pattern is replaced by or for some fixed respectively. There are by now many proofs of this theorem (see this recent paper of Lyall for a survey), but most proofs involve some form of Fourier analysis (or spectral theory). This may be compared with the standard proof of Roth’s theorem, which combines some Fourier analysis with what is now known as the density increment argument.

A few years ago, Ben Green, Tamar Ziegler, and myself observed that it is possible to prove the Furstenberg-Sarkozy theorem by just using the Cauchy-Schwarz inequality (or van der Corput lemma) and the density increment argument, removing all invocations of Fourier analysis, and instead relying on Cauchy-Schwarz to linearise the quadratic shift . As such, this theorem can be considered as even more elementary than Roth’s theorem (and its proof can be viewed as a toy model for the proof of Roth’s theorem). We ended up not doing too much with this observation, so decided to share it here.

The first step is to use the density increment argument that goes back to Roth. For any , let denote the assertion that for sufficiently large, all sets of density at least contain a pair with non-zero. Note that is vacuously true for . We will show that for any , one has the implication

for some absolute constant . This implies that is true for any (as can be seen by considering the infimum of all for which holds), which gives Theorem 1.

It remains to establish the implication (1). Suppose for sake of contradiction that we can find for which holds (for some sufficiently small absolute constant ), but fails. Thus, we can find arbitrarily large , and subsets of of density at least , such that contains no patterns of the form with non-zero. In particular, we have

(The exact ranges of and are not too important here, and could be replaced by various other small powers of if desired.)

Let be the density of , so that . Observe that

and

If we thus set , then

In particular, for large enough,

On the other hand, one easily sees that

and hence by the Cauchy-Schwarz inequality

which we can rearrange as

Shifting by we obtain (again for large enough)

In particular, by the pigeonhole principle (and deleting the diagonal case , which we can do for large enough) we can find distinct such that

so in particular

If we set and shift by , we can simplify this (again for large enough) as

On the other hand, since

we have

for any , and thus

Averaging this with (2) we conclude that

In particular, by the pigeonhole principle we can find such that

or equivalently has density at least on the arithmetic progression , which has length and spacing , for some absolute constant . By partitioning this progression into subprogressions of spacing and length (plus an error set of size , we see from the pigeonhole principle that we can find a progression of length and spacing on which has density at least (and hence at least ) for some absolute constant . If we then apply the induction hypothesis to the set

we conclude (for large enough) that contains a pair for some natural numbers with non-zero. This implies that lie in , a contradiction, establishing the implication (1).

A more careful analysis of the above argument reveals a more quantitative version of Theorem 1: for (say), any subset of of density at least for some sufficiently large absolute constant contains a pair with non-zero. This is not the best bound known; a (difficult) result of Pintz, Steiger, and Szemeredi allows the density to be as low as . On the other hand, this already improves on the (simpler) Fourier-analytic argument of Green that works for densities at least (although the original argument of Sarkozy, which is a little more intricate, works up to ). In the other direction, a construction of Rusza gives a set of density without any pairs .

Remark 1 A similar argument also applies with replaced by for fixed , because this sort of pattern is preserved by affine dilations into arithmetic progressions whose spacing is a power. By re-introducing Fourier analysis, one can also perform an argument of this type for where is the sum of two squares; see the above-mentioned paper of Green for details. However there seems to be some technical difficulty in extending it to patterns of the form for polynomials that consist of more than a single monomial (and with the normalisation , to avoid local obstructions), because one no longer has this preservation property.

Abstract algebraic analogues of integration are less well known, but can still be developed. To motivate such an abstraction, consider the integration functional from the space of complex-valued Schwarz functions to the complex numbers, defined by

where the integration on the right is the usual Lebesgue integral (or improper Riemann integral) from analysis. This functional obeys two obvious algebraic properties. Firstly, it is linear over , thus

and

for all and . Secondly, it is translation invariant, thus

for all , where is the translation of by . Motivated by the uniqueness theory of Haar measure, one might expect that these two axioms already uniquely determine after one sets a normalisation, for instance by requiring that

This is not quite true as stated (one can modify the proof of the Hahn-Banach theorem, after first applying a Fourier transform, to create pathological translation-invariant linear functionals on that are not multiples of the standard Fourier transform), but if one adds a mild analytical axiom, such as continuity of (using the usual Schwartz topology on ), then the above axioms are enough to uniquely pin down the notion of integration. Indeed, if is a continuous linear functional that is translation invariant, then from the linearity and translation invariance axioms one has

for all and non-zero reals . If is Schwartz, then as , one can verify that the Newton quotients converge in the Schwartz topology to the derivative of , so by the continuity axiom one has

Next, note that any Schwartz function of integral zero has an antiderivative which is also Schwartz, and so annihilates all zero-integral Schwartz functions, and thus must be a scalar multiple of the usual integration functional. Using the normalisation (4), we see that must therefore be the usual integration functional, giving the claimed uniqueness.

Motivated by the above discussion, we can define the notion of an abstract integration functional taking values in some vector space , and applied to inputs in some other vector space that enjoys a linear action (the “translation action”) of some group , as being a functional which is both linear and translation invariant, thus one has the axioms (1), (2), (3) for all , scalars , and . The previous discussion then considered the special case when , , , and was the usual translation action.

Once we have performed this abstraction, we can now present analogues of classical integration which bear very little analytic resemblance to the classical concept, but which still have much of the algebraic structure of integration. Consider for instance the situation in which we keep the complex range , the translation group , and the usual translation action , but we replace the space of Schwartz functions by the space of polynomials of degree at most with complex coefficients, where is a fixed natural number; note that this space is translation invariant, so it makes sense to talk about an abstract integration functional . Of course, one cannot apply traditional integration concepts to non-zero polynomials, as they are not absolutely integrable. But one can repeat the previous arguments to show that any abstract integration functional must annihilate derivatives of polynomials of degree at most :

Clearly, every polynomial of degree at most is thus annihilated by , which makes a scalar multiple of the functional that extracts the top coefficient of a polynomial, thus if one sets a normalisation

for some constant , then one has

for any polynomial . So we see that up to a normalising constant, the operation of extracting the top order coefficient of a polynomial of fixed degree serves as the analogue of integration. In particular, despite the fact that integration is supposed to be the “opposite” of differentiation (as indicated for instance by (5)), we see in this case that integration is basically (-fold) differentiation; indeed, compare (6) with the identity

In particular, we see, in contrast to the usual Lebesgue integral, the integration functional (6) can be localised to an arbitrary location: one only needs to know the germ of the polynomial at a single point in order to determine the value of the functional (6). This localisation property may initially seem at odds with the translation invariance, but the two can be reconciled thanks to the extremely rigid nature of the class , in contrast to the Schwartz class which admits bump functions and so can generate local phenomena that can only be detected in small regions of the underlying spatial domain, and which therefore forces any translation-invariant integration functional on such function classes to measure the function at every single point in space.

The reversal of the relationship between integration and differentiation is also reflected in the fact that the abstract integration operation on polynomials interacts with the scaling operation in essentially the opposite way from the classical integration operation. Indeed, for classical integration on , one has

for Schwartz functions , and so in this case the integration functional obeys the scaling law

In contrast, the abstract integration operation defined in (6) obeys the opposite scaling law

Remark 1 One way to interpret what is going on is to view the integration operation (6) as a renormalised version of integration. A polynomial is, in general, not absolutely integrable, and the partial integrals

diverge as . But if one renormalises these integrals by the factor , then one recovers convergence,

thus giving an interpretation of (6) as a renormalised classical integral, with the renormalisation being responsible for the unusual scaling relationship in (7). However, this interpretation is a little artificial, and it seems that it is best to view functionals such as (6) from an abstract algebraic perspective, rather than to try to force an analytic interpretation on them.

Now we return to the classical Lebesgue integral

As noted earlier, this integration functional has a translation invariance associated to translations along the real line , as well as a dilation invariance by real dilation parameters . However, if we refine the class of functions somewhat, we can obtain a stronger family of invariances, in which we allow complex translations and dilations. More precisely, let denote the space of all functions which are entire (or equivalently, are given by a Taylor series with an infinite radius of convergence around the origin) and also admit rapid decay in a sectorial neighbourhood of the real line, or more precisely there exists an such that for every there exists such that one has the bound

whenever . For want of a better name, we shall call elements of this space Schwartz entire functions. This is clearly a complex vector space. A typical example of a Schwartz entire function are the complex gaussians

where are complex numbers with . From the Cauchy integral formula (and its derivatives) we see that if lies in , then the restriction of to the real line lies in ; conversely, from analytic continuation we see that every function in has at most one extension in . Thus one can identify with a subspace of , and in particular the integration functional (8) is inherited by , and by abuse of notation we denote the resulting functional as also. Note, in analogy with the situation with polynomials, that this abstract integration functional is somewhat localised; one only needs to evaluate the function on the real line, rather than the entire complex plane, in order to compute . This is consistent with the rigid nature of Schwartz entire functions, as one can uniquely recover the entire function from its values on the real line by analytic continuation.

Of course, the functional remains translation invariant with respect to real translation:

However, thanks to contour shifting, we now also have translation invariance with respect to complex translation:

where of course we continue to define the translation operator for complex by the usual formula . In a similar vein, we also have the scaling law

for any , if is a complex number sufficiently close to (where “sufficiently close” depends on , and more precisely depends on the sectoral aperture parameter associated to ); again, one can verify that lies in for sufficiently close to . These invariances (which relocalise the integration functional onto other contours than the real line ) are very useful for computing integrals, and in particular for computing gaussian integrals. For instance, the complex translation invariance tells us (after shifting by ) that

when with , and then an application of the complex scaling law (and a continuity argument, observing that there is a compact path connecting to in the right half plane) gives

using the branch of on the right half-plane for which . Using the normalisation (4) we thus have

giving the usual gaussian integral formula

This is a basic illustration of the power that a large symmetry group (in this case, the complex homothety group) can bring to bear on the task of computing integrals.

One can extend this sort of analysis to higher dimensions. For any natural number , let denote the space of all functions which is jointly entire in the sense that can be expressed as a Taylor series in which is absolutely convergent for all choices of , and such that there exists an such that for any there is for which one has the bound

whenever for all , where and . Again, we call such functions Schwartz entire functions; a typical example is the function

where is an complex symmetric matrix with positive definite real part, is a vector in , and is a complex number. We can then define an abstract integration functional by integration on the real slice :

where is the usual Lebesgue measure on . By contour shifting in each of the variables separately, we see that is invariant with respect to complex translations of each of the variables, and is thus invariant under translating the joint variable by . One can also verify the scaling law

for complex matrices sufficiently close to the origin, where . This can be seen for shear transformations by Fubini’s theorem and the aforementioned translation invariance, while for diagonal transformations near the origin this can be seen from applications of one-dimensional scaling law, and the general case then follows by composition. Among other things, these laws then easily lead to the higher-dimensional generalisation

whenever is a complex symmetric matrix with positive definite real part, is a vector in , and is a complex number, basically by repeating the one-dimensional argument sketched earlier. Here, we choose the branch of for all matrices in the indicated class for which .

Now we turn to an integration functional suitable for computing complex gaussian integrals such as

where is now a complex variable

is the adjoint

is a complex matrix with positive definite Hermitian part, are column vectors in , is a complex number, and is times Lebesgue measure on . (The factors of two here turn out to be a natural normalisation, but they can be ignored on a first reading.) As we shall see later, such integrals are relevant when performing computations on the Gaussian Unitary Ensemble (GUE) in random matrix theory. Note that the integrand here is not complex analytic due to the presence of the complex conjugates. However, this can be dealt with by the trick of replacing the complex conjugate by a variable which is formally conjugate to , but which is allowed to vary independently of . More precisely, let be the space of all functions of two independent -tuples

of complex variables, which is jointly entire in all variables (in the sense defined previously, i.e. there is a joint Taylor series that is absolutely convergent for all independent choices of ), and such that there is an such that for every there is such that one has the bound

whenever . We will call such functions Schwartz analytic. Note that the integrand in (11) is Schwartz analytic when has positive definite Hermitian part, if we reinterpret as the transpose of rather than as the adjoint of in order to make the integrand entire in and . We can then define an abstract integration functional by the formula

thus can be localised to the slice of (though, as with previous functionals, one can use contour shifting to relocalise to other slices also.) One can also write this integral as

and note that the integrand here is a Schwartz entire function on , thus linking the Schwartz analytic integral with the Schwartz entire integral. Using this connection, one can verify that this functional is invariant with respect to translating and by independent shifts in (thus giving a translation symmetry), and one also has the independent dilation symmetry

for complex matrices that are sufficiently close to the identity, where . Arguing as before, we can then compute (11) as

In particular, this gives an integral representation for the determinant-reciprocal of a complex matrix with positive definite Hermitian part, in terms of gaussian expressions in which only appears linearly in the exponential:

This formula is then convenient for computing statistics such as

for random matrices drawn from the Gaussian Unitary Ensemble (GUE), and some choice of spectral parameter with ; we review this computation later in this post. By the trick of matrix differentiation of the determinant (as reviewed in this recent blog post), one can also use this method to compute matrix-valued statistics such as

However, if one restricts attention to classical integrals over real or complex (and in particular, commuting or bosonic) variables, it does not seem possible to easily eradicate the negative determinant factors in such calculations, which is unfortunate because many statistics of interest in random matrix theory, such as the expected Stieltjes transform

which is the Stieltjes transform of the density of states. However, it turns out (as I learned recently from Peter Sarnak and Tom Spencer) that it is possible to cancel out these negative determinant factors by balancing the bosonic gaussian integrals with an equal number of fermionic gaussian integrals, in which one integrates over a family of anticommuting variables. These fermionic integrals are closer in spirit to the polynomial integral (6) than to Lebesgue type integrals, and in particular obey a scaling law which is inverse to the Lebesgue scaling (in particular, a linear change of fermionic variables ends up transforming a fermionic integral by rather than ), which conveniently cancels out the reciprocal determinants in the previous calculations. Furthermore, one can combine the bosonic and fermionic integrals into a unified integration concept, known as the Berezin integral (or Grassmann integral), in which one integrates functions of supervectors (vectors with both bosonic and fermionic components), and is of particular importance in the theory of supersymmetry in physics. (The prefix “super” in physics means, roughly speaking, that the object or concept that the prefix is attached to contains both bosonic and fermionic aspects.) When one applies this unified integration concept to gaussians, this can lead to quite compact and efficient calculations (provided that one is willing to work with “super”-analogues of various concepts in classical linear algebra, such as the supertrace or superdeterminant).

Abstract integrals of the flavour of (6) arose in quantum field theory, when physicists sought to formally compute integrals of the form

where are familiar commuting (or bosonic) variables (which, in particular, can often be localised to be scalar variables taking values in or ), while were more exotic anticommuting (or fermionic) variables, taking values in some vector space of fermions. (As we shall see shortly, one can formalise these concepts by working in a supercommutative algebra.) The integrand was a formally analytic function of , in that it could be expanded as a (formal, noncommutative) power series in the variables . For functions that depend only on bosonic variables, it is certainly possible for such analytic functions to be in the Schwartz class and thus fall under the scope of the classical integral, as discussed previously. However, functions that depend on fermionic variables behave rather differently. Indeed, a fermonic variable must anticommute with itself, so that . In particular, any power series in terminates after the linear term in , so that a function can only be analytic in if it is a polynomial of degree at most in ; more generally, an analytic function of fermionic variables must be a polynomial of degree at most , and an analytic function of bosonic and fermionic variables can be Schwartz in the bosonic variables but will be polynomial in the fermonic variables. As such, to interpret the integral (14), one can use classical (Lebesgue) integration (or the variants discussed above for integrating Schwartz entire or Schwartz analytic functions) for the bosonic variables, but must use abstract integrals such as (6) for the fermonic variables, leading to the concept of Berezin integration mentioned earlier.

In this post I would like to set out some of the basic algebraic formalism of Berezin integration, particularly with regards to integration of gaussian-type expressions, and then show how this formalism can be used to perform computations involving GUE (for instance, one can compute the density of states of GUE by this machinery without recourse to the theory of orthogonal polynomials). The use of supersymmetric gaussian integrals to analyse ensembles such as GUE appears in the work of Efetov (and was also proposed in the slightly earlier works of Parisi-Sourlas and McKane, with a related approach also appearing in the work of Wegner); the material here is adapted from this survey of Mirlin, as well as the later papers of Disertori-Pinson-Spencer and of Disertori.

— 1. Grassmann algebra and Berezin integration —

Berezin integration can be performed on functions defined on the vectors in any supercommutative algebra, or even more generally on a supermanifold, but for the purposes of the applications to random matrix theory discussed here, we will only need to understand Berezin integration for analytic functions of bosonic variables and fermionic variables.

We now set up the formal mathematical framework. We will need a space of basic fermions, which can be taken to be any infinite-dimensional abstract complex vector space. The infinite dimensionality of is convenient to avoid certain degeneracies; it may seem dangerous from an analysis perspective to integrate over such spaces, but as we will be performing integration from a purely algebraic viewpoint, this will not be a concern. (Indeed, one could avoid dealing with the individual elements of space altogether, and work instead with certain rings of functions on (thus treating as a noncommutative scheme, rather than as a set of points), but we will not adopt this viewpoint here.)

We then form the -fold exterior powers , which is the universal complex vector space generated by the -fold wedge products of elements of , subject to the requirement that the wedge product is bilinear, and also antisymmetric on elements of . We then form the exterior algebra of as the direct sum of all these exterior powers. If one endows this algebra with the wedge product , one obtains a complex algebra, since the wedge product is bilinear and associative. By abuse of notation, we will write the wedge product simply as .

We split into the space of bosons (arising from exterior powers of even order) and the space of fermions (exterior powers of odd order). Thus, for instance, complex scalars (which make up ) are bosons, while elements of are fermions (i.e. basic fermions are fermions). We observe that the product of two bosons or two fermions is a boson, while the product of a boson and a fermion is a fermion, which gives the structure of a superalgebra (i.e. -graded algebra, with and being the and graded components).

Generally speaking, we will try to use Roman symbols such as to denote bosons, and Greek symbols such as to denote fermions; we will also try to use capital Greek symbols (such as ) to denote combinations of bosons and fermions.

It is easy to verify (as can be done for instance by using a basis for , with the attendant basis , for ), that bosonic elements of are central (they commute with both bosons and fermions), while fermionic elements of commute with bosonic elements but anticommute with each other. (In other words, the superalgebra is supercommutative.)

A fermionic element will commute with all bosonic elements and anticommute with fermonic elements, which in particular implies that

One corollary of this (and the anticommutativity of with itself) is that any product in which contains two copies of will necessarily vanish. Another corollary is that all elements in are nilpotent, so that for some . In particular, every element in can be decomposed as the sum of a scalar and a nilpotent (in fact, this decomposition is unique). A further corollary is the fact the algebra is locally finitely dimensional, in the sense that every finite collection of elements in generates a finite dimensional subalgebra of . Among other things, this implies that every element of can be exponentiated by the usual power series

Thus, for instance, the exponential of a bosonic element is again a bosonic element, while the exponential of a fermion is just a linear function, since anticommutes with itself and thus squares to zero:

As bosonic elements are central, we also see that we have the usual formula

whenever is bosonic and is an arbitrary element of .

We now consider functions of bosonic variables and fermionic variables . We will abbreviate as , and write

We will restrict attention to functions which are strongly analytic in the sense that they can be written as a strongly convergent noncommutative Taylor series in the variables with coefficients in . By strongly convergence, we mean that for any given choice of , all of the terms in the Taylor series lie in a finite dimensional subspace of , and the series is absolutely convergent in that finite dimensional subspace. (One could consider more relaxed notions of convergence (and thus of analyticity) here, but this strong notion of analyticity is already obeyed by the functions we will care about in applications, namely supercommutative gaussian functions with polynomial weights, so we will not need to consider more general classes of analytic functions here.)

Let denote the space of strongly analytic functions from to . This is clearly a complex algebra, and contains all the polynomials in the variables with coefficients in , as well as exponentials of such polynomials. It is also translation invariant in all of the variables (this is a variant of the basic fact in real analysis that if a Taylor series has infinite radius of convergence at the origin, then it is also equal to a Taylor series with infinite radius of sequence at any other point). On the other hand, by collecting terms in for any , we see that any strongly analytic function can be written in the form

for some strongly analytic functions . In fact, and are uniquely determined from ; is necessarily equal to , and if were not unique, then on subtraction one could find an element with the property that for all , which is not possible because is infinite dimensional.

We then define the (one-dimensional) Berezin integral

of a strongly analytic function with respect to the variable by the formula

the normalisation factor is convenient for gaussian integration calculations, as we shall see later, but can be ignored for now. This is a functional from to , which is an abstract integration functional in the sense discussed in the the introduction, because the functional is invariant with respect to translations of the variable by elements of . It also obeys the scaling law

for any invertible bosonic element , as follows immediately from the definitions.

We can iterate the above integration operation. For instance, any can be fully decomposed in terms of the fermionic variables as

where are strongly analytic functions of just the bosonic variables , and the sum ranges over tuples . We can then define the Berezin integral

of a strongly analytic function over all the fermionic variables

at once, by the formula

This is an abstract integration functional from to which is invariant under translations of by ; it can also be viewed as the iteration of the one-dimensional integrations by the Fubini-type formula

(note the reversal of the order of integration here). Much as fermions themselves anticommute with each other, one-dimensional Berezin integrals over fermonic variables also anticommute with each other, thus for instance

(compare with integration of differential forms, with ). One also verifies the scaling law

for any invertible matrix with bosonic entries, which can be verified for instance by first checking it in the case of diagonal matrices, permutation matrices, and shear matrices, and then observing that these generate all the other invertible matrices.

We can combine integration over fermionic variables with the more familiar integration over bosonic variables. We will focus attention on complex bosonic and fermionic integration rather than real bosonic and fermionic integration, as this will be the integration concept that is relevant for computations involving GUE. Thus, we will now consider strongly analytic functions of bosonic variables and fermionic variables . As previously discussed in the integration of Schwartz analytic functions, we allow the variable to vary independently of the variable despite being formally being denoted as an adjoint to , and similarly for and .

Observe that a strongly analytic function of purely bosonic variables will have all Taylor coefficients take values in a finite dimensional subspace of (otherwise it will not be strongly analytic for complex scalar non-zero ). In particular, if we restrict the bosonic variables to be complex scalars, then takes values in this subspace . We then say that is Schwartz analytic if the restriction to lies in , thus every component of this restriction lies in . Note that this restriction to is sufficient to recover the values of at all other values in , because one can read off all the Taylor coefficients of from this restriction. We denote the space of such Schwartz analytic functions as . We then use the functionals (12) to define Berezin integration on one or more pairs of bosonic variables. For instance, the Berezin integral

will, by definition, be the Lebesgue integral

recalling that is times Lebesgue measure on the complex plane in the variable, and similarly

is the quantity

One easily verifies that Berezin integration with respect to a single pair of bosonic variables maps to , and integration with respect to all the bosonic variables maps to .

As discussed in the introduction, a bosonic integral is invariant with respect to independent translations of the and by any complex shifts. It turns out that these integrals are in fact also invariant under independent translations of by arbitrary bosonic shifts. For sake of notation we will just illustrate this in the case. From the invariance under complex shifts we have

for any complex . But both sides of this equation are entire in both variables , so this identity must also hold on the level of (commutative) formal power series. Specialising from formal variables to bosonic variables we obtain the claim. For similar reasons, we have the scaling law

for all invertible matrices with bosonic entries and scalar part sufficiently close to the identity, because the claim was already shown to be true for complex entries, and both sides are analytic in .

A function of bosonic and fermionic variables and their formal adjoints will be called Schwartz analytic if each of its components under the decomposition (16) is Schwartz analytic, and the space of such functions will be denoted . One can then perform Berezin integration with respect to a pair of bosonic variables by integrating each term in (16) separately; this creates an integration functional from to . Similarly, one can integrate out all the bosonic variables at once, creating an integration functional from to . Meanwhile, fermionic integration in a pair maps can be verified to map to , and integrating out all pairs at once leads to a functional from to . Finally, one can check that bosonic integration commutes with either fermionic and bosonic integration, and fermionic integration anticommutes with fermionic integration; in particular, integrating a pair of is an operation that commutes with other such operations or with bosonic integration. Because of this, one can now define the full Berezin integral

of a Schwartz analytic function by integrating out all the pairs and (with the order in which these pairs are integrated being irrelevant). This gives an integration functional from to . From the translation invariance properties of the individual bosonic and fermonic integrals, we see that this functional is invariant with respect to independent translations of and by elements of .

Example 1 Take . If are bosons with the real scalar part of being positive, then the gaussian function

can be expanded (using the nilpotent nature of ) as

or equivalently

and this is a Schwartz analytic function on . Performing the bosonic integrals (using (13)) we then get

and then on performing the fermionic integrals we obtain

If instead one performs the fermionic integral first, one obtains

and then on performing the bosonic integrals one ends up at the same place:

Note how the parameters and scale in the opposite way in this integral.

We now derive the general scaling law for Berezin integrals

in which we scale by a matrix that suitably respects the bosonic and fermonic components of . More precisely, define an supermatrix to be a block matrix of the form

where is a matrix with bosonic entries, is an matrix with fermionic entries, is an matrix with fermionic entries, and is an matrix with bosonic entries. Observe that if is an -dimensional column supervector and is an -dimensional row supervector then

Proposition 1 (Scaling law) Let be a Schwartz analytic function, and let be a matrix. If the scalar part of is sufficiently close to the identity (or equivalently, the scalar parts of are sufficiently close to the identity), then we have

where is the superdeterminant (also known as the Berezinian) of , defined by the formula