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 objects in many other algebraic categories, 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. Splitting abelian algebras into cyclic (i.e. one-dimensional) ones, we thus see that a finite-dimensional Lie algebra is solvable if and only if it is polycylic, i.e. it can be represented by a tower of cyclic extensions.
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
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:
- (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 a special case of Weyl’s complete reducibility theorem (see Theorem 32), 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:
- 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:
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 ).
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
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.
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. This also gives an alternate way to establish Theorem 32 in the appendix.
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):
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 fact, combined with the complete reducibility of -modules (a fact which we will prove in an appendix) implies that the Jordan decomposition preserves concrete semisimple Lie algebras:
Proof: By Theorem 1, is the direct sum of commuting simple algebras. It is easy to see that if commute then the Jordan decomposition of arises from the sum of the Jordan decompositions of and separately, so we may assume without loss of generality that is simple.
Observe that if splits as the direct sum of two -invariant subspaces (so that can be viewed as a subalgebra of , and the elements of can be viewed as being block-diagonal in a suitable basis of ), then the claim for follows from that of and . So by an induction on dimension, it suffices to establish the claim under the hypothesis that is indecomposable, in that it cannot be expressed as the direct sum of two non-trivial invariant subspaces.
In the appendix we will show that every invariant subspace of is complemented in that one can write for some invariant subspace . Assuming this fact, it suffices to establish the claim in the case that is irreducible, in the sense that it contains no proper invariant subspaces.
By Lemma 7, the operation is a derivation on , thus there exists such that for all , thus centralises . By Schur’s lemma and the hypothesis of irreducibility, we conclude that is a multiple of a constant . Onthe other hand, every element of has trace zero since ; in particular, and have trace zero, and so has trace zero. But this trace is just , so we conclude that and the claim follows.
This allows us to make the Jordan decomposition universal for semisimple algebras:
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
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:
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:
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 trivial, 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:
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
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
Conversely, these representations (and their direct sums) describe (up to isomorphism) all of the representations of :
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
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 , as done in the Appendix.
— 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
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
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
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
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 .
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 .
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 :
- (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)):
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).
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 .
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 .
Remark 6 A slightly different way to reach the classification is to replace the Dynkin diagram by the extended Dynkin diagram in which one also adds the maximal negative root in addition to the simple roots; this breaks the linear independence, but one can then label each vertex by the coefficient in the linear combination needed to make the roots sum to zero, and one can then analyse these multiplicities to classify the possible diagrams and thence the root systems.
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)
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 7 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.
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 .
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 8 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 9 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
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
Lemma 30 For any roots with , one has
where are the string of roots of the form for integer .
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
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).
— 9. Appendix: Casimirs and complete reducibility —
In this appendix we supply a proof of the following fact, used in the proof of Corollary 9:
Among other things, Weyl’s complete reducibility theorem shows that every finite-dimensional linear representation of splits into the direct sum of irreducible representations, which explains the terminology. The claim is also true for semisimple Lie algebras , but we will only need the simple case here, which allows for some minor simplifications to the argument.
Proposition 33 With the hypotheses of Theorem 32, is non-degenerate.
Proof: This is a routine modification of Proposition 6 (one simply omits the use of the adjoint representation).
Once one establishes non-degeneracy, one can then define the Casimir operator by setting
whenever is a basis of and is its dual basis, thus . It is easy to see that this definition does not depend on the choice of basis, which in turn (by infinitesimally conjugating both bases by an element of the algebra ) implies that commutes with every element of .
On the other hand, does not vanish entirely. Indeed, taking traces and using (26) we see that
This already gives an important special case of Theorem 32:
Proposition 34 Theorem 32 is true when has codimension one and is irreducible.
Proof: The Lie algebra acts on the one-dimensional space ; since (from the simplicity hypothesis), we conclude that this action is trivial. In other words, each element of maps to , so the Casimir operator does as well. In particular, the trace of on is the same as the trace of on . On the other hand, by Schur’s lemma, is a constant on ; applying (27), we conclude that this constant is non-zero. Thus is non-degenerate on , but is not full rank on as it maps to . Thus it must have a one-dimensional null-space which is complementary to . As commutes with , is -invariant, and the claim follows.
We can then remove the irreducibility hypothesis:
Proposition 35 Theorem 32 is true when has codimension one.
We remark that this statement is essentially a reformulation of Whitehead’s lemma.
Proof: We induct on the dimension of (or ). If is irreducible then we are already done, so suppose that has a proper invariant subspace . Then has codimension one in , so by the induction hypothesis is complemented by a one-dimensional invariant subspace of , which lifts to an invariant subspace of in which has codimension one. By the induction hypothesis again, is complemented by a one-dimensional invariant subspace in , and it is then easy to see that also complements in , and the claim follows.
Next, we remove the codimension one hypothesis instead:
Proposition 36 Theorem 32 is true when is irreducible.
Proof: Let be the space of linear maps whose restriction to is a constant multiple of the identity, and let be the subalgebra of whose restriction to vanishes. Then are -invariant (using the Lie bracket action), and has codimension one in . Applying Proposition 35 (pushing forward to , and treating the degenerate case when vanishes separately) we see that is complemented by a one-dimensional invariant subspace of . Thus there exist that does not lie in , and which commutes with every element of . The kernel of is then an invariant complement of in , and the claim follows.