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An abstract finite-dimensional complex Lie algebra, or Lie algebra for short, is a finite-dimensional complex vector space {{\mathfrak g}} together with an anti-symmetric bilinear form {[,] = [,]_{\mathfrak g}: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}} that obeys the Jacobi identity

\displaystyle  [[x,y],z] + [[y,z],x] + [[z,x],y] = 0 \ \ \ \ \ (1)

for all {x,y,z \in {\mathfrak g}}; by anti-symmetry one can also rewrite the Jacobi identity as

\displaystyle  [x,[y,z]] = [[x,y],z] + [y,[x,z]]. \ \ \ \ \ (2)

We will usually omit the subscript from the Lie bracket {[,]_{\mathfrak g}} when this will not cause ambiguity. A homomorphism {\phi: {\mathfrak g} \rightarrow {\mathfrak h}} between two Lie algebras {{\mathfrak g},{\mathfrak h}} is a linear map that respects the Lie bracket, thus {\phi([x,y]_{\mathfrak g}) =[\phi(x),\phi(y)]_{\mathfrak h}} for all {x,y \in {\mathfrak g}}. 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 {0}; 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 {{\mathfrak{gl}}(V)} of linear transformations {x: V \rightarrow V} on a finite-dimensional complex vector space (or vector space for short) {V}, with the commutator Lie bracket {[x,y] := xy-yx}; 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 {{\mathfrak{gl}}(V)} for some vector space {V}, and similarly define a representation of a Lie algebra {{\mathfrak g}} to be a homomorphism {\rho: {\mathfrak g} \rightarrow {\mathfrak h}} into a concrete Lie algebra {{\mathfrak h}}. 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 {{\mathfrak g}} is to factor it into two simpler algebras {{\mathfrak h}, {\mathfrak k}} via a short exact sequence

\displaystyle  0 \rightarrow {\mathfrak h} \rightarrow {\mathfrak g} \rightarrow {\mathfrak k} \rightarrow 0, \ \ \ \ \ (3)

thus one has an injective homomorphism from {{\mathfrak h}} to {{\mathfrak g}} and a surjective homomorphism from {{\mathfrak g}} to {{\mathfrak k}} 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 {{\mathfrak g}} with the vector space {{\mathfrak h} \times {\mathfrak k}} equipped with a Lie bracket of the form

\displaystyle  [(t,x), (s,y)]_{\mathfrak g} = ([t,s]_{\mathfrak h} + A(t,y) - A(s,x) + B(x,y), [x,y]_{\mathfrak k}) \ \ \ \ \ (4)

for some bilinear maps {A: {\mathfrak h} \times {\mathfrak k} \rightarrow {\mathfrak h}} and {B: {\mathfrak k} \times {\mathfrak k} \rightarrow {\mathfrak h}} that obey some Jacobi-type identities which we will not record here. Understanding exactly what maps {A,B} 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 {{\mathfrak g}} can be reduced to that of understanding that of its factors {{\mathfrak k}, {\mathfrak h}}. To emphasise this, I will (perhaps idiosyncratically) express the existence of a short exact sequence (3) by the ATLAS-type notation

\displaystyle  {\mathfrak g} = {\mathfrak h} . {\mathfrak k} \ \ \ \ \ (5)

although one should caution that for given {{\mathfrak h}} and {{\mathfrak k}}, there can be multiple non-isomorphic {{\mathfrak g}} that can form a short exact sequence with {{\mathfrak h},{\mathfrak k}}, so that {{\mathfrak h} . {\mathfrak k}} is not a uniquely defined combination of {{\mathfrak h}} and {{\mathfrak k}}; one could emphasise this by writing {{\mathfrak h} ._{A,B} {\mathfrak k}} instead of {{\mathfrak h} . {\mathfrak k}}, though we will not do so here. We will refer to {{\mathfrak g}} as an extension of {{\mathfrak k}} by {{\mathfrak h}}, and read the notation (5) as “ {{\mathfrak g}} is {{\mathfrak h}}-by-{{\mathfrak k}}“; 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 {{\mathfrak k} . ({\mathfrak h} . {\mathfrak l})} is also of the form {({\mathfrak k} . {\mathfrak h}) . {\mathfrak l}}, 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 {{\mathfrak h} . {\mathfrak k}} as a (twisted) “sum” of {{\mathfrak h}} and {{\mathfrak k}} rather than a “product”; for instance, we have {{\mathfrak g} = 0 . {\mathfrak g}} and {{\mathfrak g} = {\mathfrak g} . 0}, and also {\dim {\mathfrak h} . {\mathfrak k} = \dim {\mathfrak h} + \dim {\mathfrak k}}.

Special examples of extensions {{\mathfrak h} .{\mathfrak k}} of {{\mathfrak k}} by {{\mathfrak h}} include the direct sum (or direct product) {{\mathfrak h} \oplus {\mathfrak k}} (also denoted {{\mathfrak h} \times {\mathfrak k}}), which is given by the construction (4) with {A} and {B} both vanishing, and the split extension (or semidirect product) {{\mathfrak h} : {\mathfrak k} = {\mathfrak h} :_\rho {\mathfrak k}} (also denoted {{\mathfrak h} \ltimes {\mathfrak k} = {\mathfrak h} \ltimes_\rho {\mathfrak k}}), which is given by the construction (4) with {B} vanishing and the bilinear map {A: {\mathfrak h} \times {\mathfrak k} \rightarrow {\mathfrak h}} taking the form

\displaystyle  A( t, x ) = \rho(x)(t)

for some representation {\rho: {\mathfrak k} \rightarrow \hbox{Der} {\mathfrak h}} of {{\mathfrak k}} in the concrete Lie algebra of derivations {\hbox{Der} {\mathfrak h} \subset {\mathfrak{gl}}({\mathfrak h})} of {{\mathfrak h}}, that is to say the algebra of linear maps {D: {\mathfrak h} \rightarrow {\mathfrak h}} that obey the Leibniz rule

\displaystyle  D[s,t]_{\mathfrak h} = [Ds,t]_{\mathfrak h} + [s,Dt]_{\mathfrak h}

for all {s,t \in {\mathfrak h}}. (The derivation algebra {\hbox{Der} {\mathfrak g}} of a Lie algebra {{\mathfrak g}} is analogous to the automorphism group {\hbox{Aut}(G)} of a Lie group {G}, with the two concepts being intertwined by the tangent space functor {G \mapsto {\mathfrak g}} 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 {{\mathfrak g}} as an extension {{\mathfrak h} . {\mathfrak k}} of a smaller Lie algebra {{\mathfrak k}} by another smaller Lie algebra {{\mathfrak h}}. One is to locate a Lie algebra ideal (or ideal for short) {{\mathfrak h}} in {{\mathfrak g}}, thus {[{\mathfrak h},{\mathfrak g}] \subset {\mathfrak h}}, where {[{\mathfrak h},{\mathfrak g}]} denotes the Lie algebra generated by {\{ [x,y]: x \in {\mathfrak h}, y \in {\mathfrak g} \}}, and then take {{\mathfrak k}} to be the quotient space {{\mathfrak g}/{\mathfrak h}} in the usual manner; one can check that {{\mathfrak h}}, {{\mathfrak k}} are also Lie algebras and that we do indeed have a short exact sequence

\displaystyle  {\mathfrak g} = {\mathfrak h} . ({\mathfrak g}/{\mathfrak h}).

Conversely, whenever one has a factorisation {{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}, one can identify {{\mathfrak h}} with an ideal in {{\mathfrak g}}, and {{\mathfrak k}} with the quotient of {{\mathfrak g}} by {{\mathfrak h}}.

The other general way to obtain such a factorisation is is to start with a homomorphism {\rho: {\mathfrak g} \rightarrow {\mathfrak m}} of {{\mathfrak g}} into another Lie algebra {{\mathfrak m}}, take {{\mathfrak k}} to be the image {\rho({\mathfrak g})} of {{\mathfrak g}}, and {{\mathfrak h}} to be the kernel {\hbox{ker} \rho := \{ x \in {\mathfrak g}: \rho(x) = 0 \}}. Again, it is easy to see that this does indeed create a short exact sequence:

\displaystyle  {\mathfrak g} = \hbox{ker} \rho . \rho({\mathfrak g}).

Conversely, whenever one has a factorisation {{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}, one can identify {{\mathfrak k}} with the image of {{\mathfrak g}} under some homomorphism, and {{\mathfrak h}} with the kernel of that homomorphism. Note that if a representation {\rho: {\mathfrak g} \rightarrow {\mathfrak m}} is faithful (i.e. injective), then the kernel is trivial and {{\mathfrak g}} is isomorphic to {\rho({\mathfrak g})}.

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 {{\mathfrak g}}, in which the Lie bracket identically vanishes. Every one-dimensional Lie algebra is automatically abelian, and thus isomorphic to the scalar algebra {{\bf C}}. Conversely, by using an arbitrary linear basis of {{\mathfrak g}}, 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 {{\bf C}}.

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

\displaystyle  {\mathfrak g} = [{\mathfrak g},{\mathfrak g}] . ({\mathfrak g}/[{\mathfrak g},{\mathfrak g}]).

The algebra {{\mathfrak g}/[{\mathfrak g},{\mathfrak g}]} is the maximal abelian quotient of {{\mathfrak g}}, and is known as the abelianisation of {{\mathfrak g}}. 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 {{\mathfrak g}}. From this, it is natural to associate two series to any Lie algebra {{\mathfrak g}}, the lower central series

\displaystyle  {\mathfrak g}_1 = {\mathfrak g}; {\mathfrak g}_2 := [{\mathfrak g}, {\mathfrak g}_1]; {\mathfrak g}_3 := [{\mathfrak g}, {\mathfrak g}_2]; \ldots

and the derived series

\displaystyle  {\mathfrak g}^{(1)} := {\mathfrak g}; {\mathfrak g}^{(2)} := [{\mathfrak g}^{(1)}, {\mathfrak g}^{(1)}]; {\mathfrak g}^{(3)} := [{\mathfrak g}^{(2)}, {\mathfrak g}^{(2)}]; \ldots.

By induction we see that these are both decreasing series of ideals of {{\mathfrak g}}, with the derived series being slightly smaller ({{\mathfrak g}^{(k)} \subseteq {\mathfrak g}_k} for all {k}). 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 {{\mathfrak{gl}}_n = {\mathfrak{gl}}({\bf C}^n)}, which can be identified with the Lie algebra of {n \times n} complex matrices, the subalgebra {{\mathfrak n}} of strictly upper triangular matrices is nilpotent (but not abelian for {n \geq 3}), while the subalgebra {{\mathfrak n}} of upper triangular matrices is solvable (but not nilpotent for {n \geq 2}). 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

\displaystyle  {\mathfrak g} = {\mathfrak a}_1 . ({\mathfrak a}_2 . \ldots ({\mathfrak a}_{k-1} . {\mathfrak a}_k) \ldots )

for some abelian {{\mathfrak a}_1,\ldots,{\mathfrak a}_k}. Similarly, a Lie algebra {{\mathfrak g}} is nilpotent if it is expressible as a tower of central extensions (so that in all the extensions {{\mathfrak h} . {\mathfrak k}} in the above factorisation, {{\mathfrak h}} is central in {{\mathfrak h} . {\mathfrak k}}, where we say that {{\mathfrak h}} is central in {{\mathfrak g}} if {[{\mathfrak h},{\mathfrak g}]=0}). We also see that an extension {{\mathfrak h} . {\mathfrak k}} is solvable if and only of both factors {{\mathfrak h}, {\mathfrak k}} 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 {{\mathfrak g}} has an adjoint representation {\hbox{ad}: {\mathfrak g} \rightarrow \hbox{ad} {\mathfrak g} \subset {\mathfrak{gl}}({\mathfrak g})}, where for each {x \in {\mathfrak g}}, {\hbox{ad} x \in {\mathfrak{gl}}({\mathfrak g})} is the linear map {(\hbox{ad} x)(y) := [x,y]}; one easily verifies that this is indeed a representation (indeed, (2) is equivalent to the assertion that {\hbox{ad} [x,y] = [\hbox{ad} x, \hbox{ad} y]} for all {x,y \in {\mathfrak g}}). The kernel of this representation is the center {Z({\mathfrak g}) := \{ x \in {\mathfrak g}: [x,{\mathfrak g}] = 0\}}, which the maximal central subalgebra of {{\mathfrak g}}. We thus have the short exact sequence

\displaystyle  {\mathfrak g} = Z({\mathfrak g}) . \hbox{ad} g \ \ \ \ \ (6)

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 {{\mathfrak g}} is simple if it is non-abelian and has no ideals other than {0} and {{\mathfrak g}}; thus simple Lie algebras cannot be factored {{\mathfrak g} = {\mathfrak h} . {\mathfrak k}} into strictly smaller algebras {{\mathfrak h},{\mathfrak k}}. In particular, simple Lie algebras are automatically perfect and centerless. We have the following fundamental theorem:

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

  • (i) {{\mathfrak g}} does not contain any non-trivial solvable ideal.
  • (ii) {{\mathfrak g}} does not contain any non-trivial abelian ideal.
  • (iii) The Killing form {K: {\mathfrak g} \times {\mathfrak g} \rightarrow {\bf C}}, defined as the bilinear form {K(x,y) := \hbox{tr}_{\mathfrak g}( (\hbox{ad} x) (\hbox{ad} y) )}, is non-degenerate on {{\mathfrak g}}.
  • (iv) {{\mathfrak g}} 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 {{\mathfrak h}} and {{\mathfrak k}} are solvable ideals of a Lie algebra {{\mathfrak g}}, then it is not difficult to see that the vector sum {{\mathfrak h}+{\mathfrak k}} is also a solvable ideal (because on quotienting by {{\mathfrak h}} we see that the derived series of {{\mathfrak h}+{\mathfrak k}} must eventually fall inside {{\mathfrak h}}, and thence must eventually become trivial by the solvability of {{\mathfrak h}}). As our Lie algebras are finite dimensional, we conclude that {{\mathfrak g}} has a unique maximal solvable ideal, known as the radical {\hbox{rad} {\mathfrak g}} of {{\mathfrak g}}. The quotient {{\mathfrak g}/\hbox{rad} {\mathfrak g}} is then a Lie algebra with trivial radical, and is thus semisimple by the above theorem, giving the Levi decomposition

\displaystyle  {\mathfrak g} = \hbox{rad} {\mathfrak g} . ({\mathfrak g} / \hbox{rad} {\mathfrak g})

expressing an arbitrary Lie algebra as an extension of a semisimple Lie algebra {{\mathfrak g}/\hbox{rad}{\mathfrak g}} by a solvable algebra {\hbox{rad} {\mathfrak g}} (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

\displaystyle  {\mathfrak g} = \hbox{rad} {\mathfrak g} : ({\mathfrak g} / \hbox{rad} {\mathfrak g})

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 {{\bf C}}. In principle, this means that one can understand an arbitrary Lie algebra once one understands all the simple Lie algebras (which, being defined over {{\bf C}}, 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:

  • {A_n = \mathfrak{sl}_{n+1}} for some {n \geq 1}.
  • {B_n = \mathfrak{so}_{2n+1}} for some {n \geq 2}.
  • {C_n = \mathfrak{sp}_{2n}} for some {n \geq 3}.
  • {D_n = \mathfrak{so}_{2n}} for some {n \geq 4}.
  • {E_6, E_7}, or {E_8}.
  • {F_4}.
  • {G_2}.

(The precise definition of the classical Lie algebras {A_n,B_n,C_n,D_n} and the exceptional Lie algebras {E_6,E_7,E_8,F_4,G_2} will be recalled later.)

(One can extend the families {A_n,B_n,C_n,D_n} of classical Lie algebras a little bit to smaller values of {n}, 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 {{\bf R}}), 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.

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Let {{\mathfrak g}} be a finite-dimensional Lie algebra (over the reals). Given two sufficiently small elements {x, y} of {{\mathfrak g}}, define the right Baker-Campbell-Hausdorff-Dynkin law

\displaystyle  R_y(x) := x + \int_0^1 F_R( \hbox{Ad}_x \hbox{Ad}_{ty} ) y \ dt \ \ \ \ \ (1)

where {\hbox{Ad}_x := \exp(\hbox{ad}_x)}, {\hbox{ad}_x: {\mathfrak g} \rightarrow {\mathfrak g}} is the adjoint map {\hbox{ad}_x(y) := [x,y]}, and {F_R} is the function {F_R(z) := \frac{z \log z}{z-1}}, which is analytic for {z} near {1}. Similarly, define the left Baker-Campbell-Hausdorff-Dynkin law

\displaystyle  L_x(y) := y + \int_0^1 F_L( \hbox{Ad}_{tx} \hbox{Ad}_y ) x\ dt \ \ \ \ \ (2)

where {F_L(z) := \frac{\log z}{z-1}}. One easily verifies that these expressions are well-defined (and depend smoothly on {x} and {y}) when {x} and {y} are sufficiently small.

We have the famous Baker-Campbell-Hausdoff-Dynkin formula:

Theorem 1 (BCH formula) Let {G} be a finite-dimensional Lie group over the reals with Lie algebra {{\mathfrak g}}. Let {\log} be a local inverse of the exponential map {\exp: {\mathfrak g} \rightarrow G}, defined in a neighbourhood of the identity. Then for sufficiently small {x, y \in {\mathfrak g}}, one has

\displaystyle  \log( \exp(x) \exp(y) ) = R_y(x) = L_x(y).

See for instance these notes of mine for a proof of this formula (it is for {R_y}, but one easily obtains a similar proof for {L_x}).

In particular, one can give a neighbourhood of the identity in {{\mathfrak g}} the structure of a local Lie group by defining the group operation {\ast} as

\displaystyle  x \ast y := R_y(x) = L_x(y) \ \ \ \ \ (3)

for sufficiently small {x, y}, and the inverse operation by {x^{-1} := -x} (one easily verifies that {R_x(-x) = L_x(-x) = 0} for all small {x}).

It is tempting to reverse the BCH formula and conclude (the local form of) Lie’s third theorem, that every finite-dimensional Lie algebra is isomorphic to the Lie algebra of some local Lie group, by using (3) to define a smooth local group structure on a neighbourhood of the identity. (See this previous post for a definition of a local Lie group.) The main difficulty in doing so is in verifying that the definition (3) is well-defined (i.e. that {R_y(x)} is always equal to {L_x(y)}) and locally associative. The well-definedness issue can be trivially disposed of by using just one of the expressions {R_y(x)} or {L_x(y)} as the definition of {\ast} (though, as we shall see, it will be very convenient to use both of them simultaneously). However, the associativity is not obvious at all.

With the assistance of Ado’s theorem, which places {{\mathfrak g}} inside the general linear Lie algebra {\mathfrak{gl}_n({\bf R})} for some {n}, one can deduce both the well-definedness and associativity of (3) from the Baker-Campbell-Hausdorff formula for {\mathfrak{gl}_n({\bf R})}. However, Ado’s theorem is rather difficult to prove (see for instance this previous blog post for a proof), and it is natural to ask whether there is a way to establish these facts without Ado’s theorem.

After playing around with this for some time, I managed to extract a direct proof of well-definedness and local associativity of (3), giving a proof of Lie’s third theorem independent of Ado’s theorem. This is not a new result by any means, (indeed, the original proofs of Lie and Cartan of Lie’s third theorem did not use Ado’s theorem), but I found it an instructive exercise to work out the details, and so I am putting it up on this blog in case anyone else is interested (and also because I want to be able to find the argument again if I ever need it in the future).

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Hilbert’s fifth problem concerns the minimal hypotheses one needs to place on a topological group {G} to ensure that it is actually a Lie group. In the previous set of notes, we saw that one could reduce the regularity hypothesis imposed on {G} to a “{C^{1,1}}” condition, namely that there was an open neighbourhood of {G} that was isomorphic (as a local group) to an open subset {V} of a Euclidean space {{\bf R}^d} with identity element {0}, and with group operation {\ast} obeying the asymptotic

\displaystyle  x \ast y = x + y + O(|x| |y|)

for sufficiently small {x,y}. We will call such local groups {(V,\ast)} {C^{1,1}} local groups.

We now reduce the regularity hypothesis further, to one in which there is no explicit Euclidean space that is initially attached to {G}. Of course, Lie groups are still locally Euclidean, so if the hypotheses on {G} do not involve any explicit Euclidean spaces, then one must somehow build such spaces from other structures. One way to do so is to exploit an ambient space with Euclidean or Lie structure that {G} is embedded or immersed in. A trivial example of this is provided by the following basic fact from linear algebra:

Lemma 1 If {V} is a finite-dimensional vector space (i.e. it is isomorphic to {{\bf R}^d} for some {d}), and {W} is a linear subspace of {V}, then {W} is also a finite-dimensional vector space.

We will establish a non-linear version of this statement, known as Cartan’s theorem. Recall that a subset {S} of a {d}-dimensional smooth manifold {M} is a {d'}-dimensional smooth (embedded) submanifold of {M} for some {0 \leq d' \leq d} if for every point {x \in S} there is a smooth coordinate chart {\phi: U \rightarrow V} of a neighbourhood {U} of {x} in {M} that maps {x} to {0}, such that {\phi(U \cap S) = V \cap {\bf R}^{d'}}, where we identify {{\bf R}^{d'} \equiv {\bf R}^{d'} \times \{0\}^{d-d'}} with a subspace of {{\bf R}^d}. Informally, {S} locally sits inside {M} the same way that {{\bf R}^{d'}} sits inside {{\bf R}^d}.

Theorem 2 (Cartan’s theorem) If {H} is a (topologically) closed subgroup of a Lie group {G}, then {H} is a smooth submanifold of {G}, and is thus also a Lie group.

Note that the hypothesis that {H} is closed is essential; for instance, the rationals {{\bf Q}} are a subgroup of the (additive) group of reals {{\bf R}}, but the former is not a Lie group even though the latter is.

Exercise 1 Let {H} be a subgroup of a locally compact group {G}. Show that {H} is closed in {G} if and only if it is locally compact.

A variant of the above results is provided by using (faithful) representations instead of embeddings. Again, the linear version is trivial:

Lemma 3 If {V} is a finite-dimensional vector space, and {W} is another vector space with an injective linear transformation {\rho: W \rightarrow V} from {W} to {V}, then {W} is also a finite-dimensional vector space.

Here is the non-linear version:

Theorem 4 (von Neumann’s theorem) If {G} is a Lie group, and {H} is a locally compact group with an injective continuous homomorphism {\rho: H \rightarrow G}, then {H} also has the structure of a Lie group.

Actually, it will suffice for the homomorphism {\rho} to be locally injective rather than injective; related to this, von Neumann’s theorem localises to the case when {H} is a local group rather a group. The requirement that {H} be locally compact is necessary, for much the same reason that the requirement that {H} be closed was necessary in Cartan’s theorem.

Example 1 Let {G = ({\bf R}/{\bf Z})^2} be the two-dimensional torus, let {H = {\bf R}}, and let {\rho: H \rightarrow G} be the map {\rho(x) := (x,\alpha x)}, where {\alpha \in {\bf R}} is a fixed real number. Then {\rho} is a continuous homomorphism which is locally injective, and is even globally injective if {\alpha} is irrational, and so Theorem 4 is consistent with the fact that {H} is a Lie group. On the other hand, note that when {\alpha} is irrational, then {\rho(H)} is not closed; and so Theorem 4 does not follow immediately from Theorem 2 in this case. (We will see, though, that Theorem 4 follows from a local version of Theorem 2.)

As a corollary of Theorem 4, we observe that any locally compact Hausdorff group {H} with a faithful linear representation, i.e. a continuous injective homomorphism from {H} into a linear group such as {GL_n({\bf R})} or {GL_n({\bf C})}, is necessarily a Lie group. This suggests a representation-theoretic approach to Hilbert’s fifth problem. While this approach does not seem to readily solve the entire problem, it can be used to establish a number of important special cases with a well-understood representation theory, such as the compact case or the abelian case (for which the requisite representation theory is given by the Peter-Weyl theorem and Pontryagin duality respectively). We will discuss these cases further in later notes.

In all of these cases, one is not really building up Euclidean or Lie structure completely from scratch, because there is already a Euclidean or Lie structure present in another object in the hypotheses. Now we turn to results that can create such structure assuming only what is ostensibly a weaker amount of structure. In the linear case, one example of this is is the following classical result in the theory of topological vector spaces.

Theorem 5 Let {V} be a locally compact Hausdorff topological vector space. Then {V} is isomorphic (as a topological vector space) to {{\bf R}^d} for some finite {d}.

Remark 1 The Banach-Alaoglu theorem asserts that in a normed vector space {V}, the closed unit ball in the dual space {V^*} is always compact in the weak-* topology. Of course, this dual space {V^*} may be infinite-dimensional. This however does not contradict the above theorem, because the closed unit ball is not a neighbourhood of the origin in the weak-* topology (it is only a neighbourhood with respect to the strong topology).

The full non-linear analogue of this theorem would be the Gleason-Yamabe theorem, which we are not yet ready to prove in this set of notes. However, by using methods similar to that used to prove Cartan’s theorem and von Neumann’s theorem, one can obtain a partial non-linear analogue which requires an additional hypothesis of a special type of metric, which we will call a Gleason metric:

Definition 6 Let {G} be a topological group. A Gleason metric on {G} is a left-invariant metric {d: G \times G \rightarrow {\bf R}^+} which generates the topology on {G} and obeys the following properties for some constant {C>0}, writing {\|g\|} for {d(g,\hbox{id})}:

  • (Escape property) If {g \in G} and {n \geq 1} is such that {n \|g\| \leq \frac{1}{C}}, then {\|g^n\| \geq \frac{1}{C} n \|g\|}.
  • (Commutator estimate) If {g, h \in G} are such that {\|g\|, \|h\| \leq \frac{1}{C}}, then

    \displaystyle  \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)

    where {[g,h] := g^{-1}h^{-1}gh} is the commutator of {g} and {h}.

Exercise 2 Let {G} be a topological group that contains a neighbourhood of the identity isomorphic to a {C^{1,1}} local group. Show that {G} admits at least one Gleason metric.

Theorem 7 (Building Lie structure from Gleason metrics) Let {G} be a locally compact group that has a Gleason metric. Then {G} is isomorphic to a Lie group.

We will rely on Theorem 7 to solve Hilbert’s fifth problem; this theorem reduces the task of establishing Lie structure on a locally compact group to that of building a metric with suitable properties. Thus, much of the remainder of the solution of Hilbert’s fifth problem will now be focused on the problem of how to construct good metrics on a locally compact group.

In all of the above results, a key idea is to use one-parameter subgroups to convert from the nonlinear setting to the linear setting. Recall from the previous notes that in a Lie group {G}, the one-parameter subgroups are in one-to-one correspondence with the elements of the Lie algebra {{\mathfrak g}}, which is a vector space. In a general topological group {G}, the concept of a one-parameter subgroup (i.e. a continuous homomorphism from {{\bf R}} to {G}) still makes sense; the main difficulties are then to show that the space of such subgroups continues to form a vector space, and that the associated exponential map {\exp: \phi \mapsto \phi(1)} is still a local homeomorphism near the origin.

Exercise 3 The purpose of this exercise is to illustrate the perspective that a topological group can be viewed as a non-linear analogue of a vector space. Let {G, H} be locally compact groups. For technical reasons we assume that {G, H} are both {\sigma}-compact and metrisable.

  • (i) (Open mapping theorem) Show that if {\phi: G \rightarrow H} is a continuous homomorphism which is surjective, then it is open (i.e. the image of open sets is open). (Hint: mimic the proof of the open mapping theorem for Banach spaces, as discussed for instance in these notes. In particular, take advantage of the Baire category theorem.)
  • (ii) (Closed graph theorem) Show that if a homomorphism {\phi: G \rightarrow H} is closed (i.e. its graph {\{ (g, \phi(g)): g \in G \}} is a closed subset of {G \times H}), then it is continuous. (Hint: mimic the derivation of the closed graph theorem from the open mapping theorem in the Banach space case, as again discussed in these notes.)
  • (iii) Let {\phi: G \rightarrow H} be a homomorphism, and let {\rho: H \rightarrow K} be a continuous injective homomorphism into another Hausdorff topological group {K}. Show that {\phi} is continuous if and only if {\rho \circ \phi} is continuous.
  • (iv) Relax the condition of metrisability to that of being Hausdorff. (Hint: Now one cannot use the Baire category theorem for metric spaces; but there is an analogue of this theorem for locally compact Hausdorff spaces.)

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In this set of notes, we describe the basic analytic structure theory of Lie groups, by relating them to the simpler concept of a Lie algebra. Roughly speaking, the Lie algebra encodes the “infinitesimal” structure of a Lie group, but is a simpler object, being a vector space rather than a nonlinear manifold. Nevertheless, thanks to the fundamental theorems of Lie, the Lie algebra can be used to reconstruct the Lie group (at a local level, at least), by means of the exponential map and the Baker-Campbell-Hausdorff formula. As such, the local theory of Lie groups is completely described (in principle, at least) by the theory of Lie algebras, which leads to a number of useful consequences, such as the following:

  • (Local Lie implies Lie) A topological group {G} is Lie (i.e. it is isomorphic to a Lie group) if and only if it is locally Lie (i.e. the group operations are smooth near the origin).
  • (Uniqueness of Lie structure) A topological group has at most one smooth structure on it that makes it Lie.
  • (Weak regularity implies strong regularity, I) Lie groups are automatically real analytic. (In fact one only needs a “local {C^{1,1}}” regularity on the group structure to obtain real analyticity.)
  • (Weak regularity implies strong regularity, II) A continuous homomorphism from one Lie group to another is automatically smooth (and real analytic).

The connection between Lie groups and Lie algebras also highlights the role of one-parameter subgroups of a topological group, which will play a central role in the solution of Hilbert’s fifth problem.

We note that there is also a very important algebraic structure theory of Lie groups and Lie algebras, in which the Lie algebra is split into solvable and semisimple components, with the latter being decomposed further into simple components, which can then be completely classified using Dynkin diagrams. This classification is of fundamental importance in many areas of mathematics (e.g. representation theory, arithmetic geometry, and group theory), and many of the deeper facts about Lie groups and Lie algebras are proven via this classification (although in such cases it can be of interest to also find alternate proofs that avoid the classification). However, it turns out that we will not need this theory in this course, and so we will not discuss it further here (though it can of course be found in any graduate text on Lie groups and Lie algebras).

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One of the fundamental structures in modern mathematics is that of a group. Formally, a group is a set {G = (G,1,\cdot,()^{-1})} equipped with an identity element {1 = 1_G \in G}, a multiplication operation {\cdot: G \times G \rightarrow G}, and an inversion operation {()^{-1}: G \rightarrow G} obeying the following axioms:

  • (Closure) If {g, h \in G}, then {g \cdot h} and {g^{-1}} are well-defined and lie in {G}. (This axiom is redundant from the above description, but we include it for emphasis.)
  • (Associativity) If {g, h, k \in G}, then {(g \cdot h) \cdot k = g \cdot (h \cdot k)}.
  • (Identity) If {g \in G}, then {g \cdot 1 = 1 \cdot g = g}.
  • (Inverse) If {g \in G}, then {g \cdot g^{-1} = g^{-1} \cdot g = 1}.

One can also consider additive groups {G = (G,0,+,-)} instead of multiplicative groups, with the obvious changes of notation. By convention, additive groups are always understood to be abelian, so it is convenient to use additive notation when one wishes to emphasise the abelian nature of the group structure. As usual, we often abbreviate {g \cdot h} by {gh} (and {1_G} by {1}) when there is no chance of confusion.

If furthermore {G} is equipped with a topology, and the group operations {\cdot, ()^{-1}} are continuous in this topology, then {G} is a topological group. Any group can be made into a topological group by imposing the discrete topology, but there are many more interesting examples of topological groups, such as Lie groups, in which {G} is not just a topological space, but is in fact a smooth manifold (and the group operations are not merely continuous, but also smooth).

There are many naturally occuring group-like objects that obey some, but not all, of the axioms. For instance, monoids are required to obey the closure, associativity, and identity axioms, but not the inverse axiom. If we also drop the identity axiom, we end up with a semigroup. Groupoids do not necessarily obey the closure axiom, but obey (versions of) the associativity, identity, and inverse axioms. And so forth.

Another group-like concept is that of a local topological group (or local group, for short), which is essentially a topological group with the closure axiom omitted (but do not obey the same axioms set as groupoids); they arise primarily in the study of local properties of (global) topological groups, and also in the study of approximate groups in additive combinatorics. Formally, a local group {G = (G, \Omega, \Lambda, 1, \cdot, ()^{-1})} is a topological space {G} equipped with an identity element {1 \in G}, a partially defined but continuous multiplication operation {\cdot: \Omega \rightarrow G} for some domain {\Omega \subset G \times G}, and a partially defined but continuous inversion operation {()^{-1}: \Lambda \rightarrow G}, where {\Lambda \subset G}, obeying the following axioms:

  • (Local closure) {\Omega} is an open neighbourhood of {G \times \{1\} \cup \{1\} \times G}, and {\Lambda} is an open neighbourhood of {1}.
  • (Local associativity) If {g, h, k \in G} are such that {(g \cdot h) \cdot k} and {g \cdot (h \cdot k)} are both well-defined, then they are equal. (Note however that it may be possible for one of these products to be defined but not the other, in contrast for instance with groupoids.)
  • (Identity) For all {g \in G}, {g \cdot 1 = 1 \cdot g = g}.
  • (Local inverse) If {g \in G} and {g^{-1}} is well-defined, then {g \cdot g^{-1} = g^{-1} \cdot g = 1}. (In particular this, together with the other axioms, forces {1^{-1} = 1}.)

We will often refer to ordinary groups as global groups (and topological groups as global topological groups) to distinguish them from local groups. Every global topological group is a local group, but not conversely.

One can consider discrete local groups, in which the topology is the discrete topology; in this case, the openness and continuity axioms in the definition are automatic and can be omitted. At the other extreme, one can consider local Lie groups, in which the local group {G} has the structure of a smooth manifold, and the group operations are smooth. We can also consider symmetric local groups, in which {\Lambda=G} (i.e. inverses are always defined). Symmetric local groups have the advantage of local homogeneity: given any {g \in G}, the operation of left-multiplication {x \mapsto gx} is locally inverted by {x \mapsto g^{-1} x} near the identity, thus giving a homeomorphism between a neighbourhood of {g} and a neighbourhood of the identity; in particular, we see that given any two group elements {g, h} in a symmetric local group {G}, there is a homeomorphism between a neighbourhood of {g} and a neighbourhood of {h}. (If the symmetric local group is also Lie, then these homeomorphisms are in fact diffeomorphisms.) This local homogeneity already simplifies a lot of the possible topology of symmetric local groups, as it basically means that the local topological structure of such groups is determined by the local structure at the origin. (For instance, all connected components of a local Lie group necessarily have the same dimension.) It is easy to see that any local group has at least one symmetric open neighbourhood of the identity, so in many situations we can restrict to the symmetric case without much loss of generality.

A prime example of a local group can be formed by restricting any global topological group {G} to an open neighbourhood {U \subset G} of the identity, with the domains

\displaystyle  \Omega := \{ (g,h) \in U: g \cdot h \in U \}

and

\displaystyle  \Lambda := \{ g \in U: g^{-1} \in U \};

one easily verifies that this gives {U} the structure of a local group (which we will sometimes call {G\downharpoonright_U} to emphasise the original group {G}). If {U} is symmetric (i.e. {U^{-1}=U}), then we in fact have a symmetric local group. One can also restrict local groups {G} to open neighbourhoods {U} to obtain a smaller local group {G\downharpoonright_U} by the same procedure (adopting the convention that statements such as {g \cdot h \in U} or {g^{-1} \in U} are considered false if the left-hand side is undefined). (Note though that if one restricts to non-open neighbourhoods of the identity, then one usually does not get a local group; for instance {[-1,1]} is not a local group (why?).)

Finite subsets of (Hausdorff) groups containing the identity can be viewed as local groups. This point of view turns out to be particularly useful for studying approximate groups in additive combinatorics, a point which I hope to expound more on later. Thus, for instance, the discrete interval {\{-9,\ldots,9\} \subset {\bf Z}} is an additive symmetric local group, which informally might model an adding machine that can only handle (signed) one-digit numbers. More generally, one can view a local group as an object that behaves like a group near the identity, but for which the group laws (and in particular, the closure axiom) can start breaking down once one moves far enough away from the identity.

One can formalise this intuition as follows. Let us say that a word {g_1 \ldots g_n} in a local group {G} is well-defined in {G} (or well-defined, for short) if every possible way of associating this word using parentheses is well-defined from applying the product operation. For instance, in order for {abcd} to be well-defined, {((ab)c)d}, {(a(bc))d}, {(ab)(cd)}, {a(b(cd))}, and {a((bc)d)} must all be well-defined. In the preceding example {\{-9,\ldots,9\}}, {-2+6+5} is not well-defined because one of the ways of associating this sum, namely {-2+(6+5)}, is not well-defined (even though {(-2+6)+5} is well-defined).

Exercise 1 (Iterating the associative law)

  • Show that if a word {g_1 \ldots g_n} in a local group is well-defined, then all ways of associating this word give the same answer, and so we can uniquely evaluate {g_1 \ldots g_n} as an element in {G}.
  • Give an example of a word {g_1 \ldots g_n} in a local group which has two ways of being associated that are both well-defined, but give different answers. (Hint: the local associativity axiom prevents this from happening for {n \leq 3}, so try {n=4}. A small discrete local group will already suffice to give a counterexample; verifying the local group axioms are easier if one makes the domain of definition of the group operations as small as one can get away with while still having the counterexample.)

Exercise 2 Show that the number of ways to associate a word {g_1 \ldots g_n} is given by the Catalan number {C_{n-1} := \frac{1}{n} \binom{2n-2}{n-1}}.

Exercise 3 Let {G} be a local group, and let {m \geq 1} be an integer. Show that there exists a symmetric open neighbourhood {U_m} of the identity such that every word of length {m} in {U_m} is well-defined in {G} (or more succinctly, {U_m^m} is well-defined). (Note though that these words will usually only take values in {G}, rather than in {U_m}, and also the sets {U_m} tend to become smaller as {m} increases.)

In many situations (such as when one is investigating the local structure of a global group) one is only interested in the local properties of a (local or global) group. We can formalise this by the following definition. Let us call two local groups {G = (G, \Omega, \Lambda, 1_G, \cdot, ()^{-1})} and {G' = (G', \Omega', \Lambda', 1_{G'}, \cdot, ()^{-1})} locally identical if they have a common restriction, thus there exists a set {U \subset G \cap G'} such that {G\downharpoonright_U = G'\downharpoonright_U} (thus, {1_G = 1_{G'}}, and the topology and group operations of {G} and {G'} agree on {U}). This is easily seen to be an equivalence relation. We call an equivalence class {[G]} of local groups a group germ.

Let {{\mathcal P}} be a property of a local group (e.g. abelianness, connectedness, compactness, etc.). We call a group germ locally {{\mathcal P}} if every local group in that germ has a restriction that obeys {{\mathcal P}}; we call a local or global group {G} locally {{\mathcal P}} if its germ is locally {{\mathcal P}} (or equivalently, every open neighbourhood of the identity in {G} contains a further neighbourhood that obeys {{\mathcal P}}). Thus, the study of local properties of (local or global) groups is subsumed by the study of group germs.

Exercise 4

  • Show that the above general definition is consistent with the usual definitions of the properties “connected” and “locally connected” from point-set topology.
  • Strictly speaking, the above definition is not consistent with the usual definitions of the properties “compact” and “local compact” from point-set topology because in the definition of local compactness, the compact neighbourhoods are certainly not required to be open. Show however that the point-set topology notion of “locally compact” is equivalent, using the above conventions, to the notion of “locally precompact inside of an ambient local group”. Of course, this is a much more clumsy terminology, and so we shall abuse notation slightly and continue to use the standard terminology “locally compact” even though it is, strictly speaking, not compatible with the above general convention.
  • Show that a local group is discrete if and only if it is locally trivial.
  • Show that a connected global group is abelian if and only if it is locally abelian. (Hint: in a connected global group, the only open subgroup is the whole group.)
  • Show that a global topological group is first-countable if and only if it is locally first countable. (By the Birkhoff-Kakutani theorem, this implies that such groups are metrisable if and only if they are locally metrisable.)
  • Let {p} be a prime. Show that the solenoid group {{\bf Z}_p \times {\bf R} / {\bf Z}^\Delta}, where {{\bf Z}_p} is the {p}-adic integers and {{\bf Z}^\Delta := \{ (n,n): n \in {\bf Z}\}} is the diagonal embedding of {{\bf Z}} inside {{\bf Z}_p \times {\bf R}}, is connected but not locally connected.

Remark 1 One can also study the local properties of groups using nonstandard analysis. Instead of group germs, one works (at least in the case when {G} is first countable) with the monad {o(G)} of the identity element {1_G} of {G}, defined as the nonstandard group elements {g = \lim_{n \rightarrow \alpha} g_n} in {{}^* G} that are infinitesimally close to the origin in the sense that they lie in every standard neighbourhood of the identity. The monad {o(G)} is closely related to the group germ {[G]}, but has the advantage of being a genuine (global) group, as opposed to an equivalence class of local groups. It is possible to recast most of the results here in this nonstandard formulation; see e.g. the classic text of Robinson. However, we will not adopt this perspective here.

A useful fact to know is that Lie structure is local. Call a (global or local) topological group Lie if it can be given the structure of a (global or local) Lie group.

Lemma 1 (Lie is a local property) A global topological group {G} is Lie if and only if it is locally Lie. The same statement holds for local groups {G} as long as they are symmetric.

We sketch a proof of this lemma below the fold. One direction is obvious, as the restriction a global Lie group to an open neighbourhood of the origin is clearly a local Lie group; for instance, the continuous interval {(-10,10) \subset {\bf R}} is a symmetric local Lie group. The converse direction is almost as easy, but (because we are not assuming {G} to be connected) requires one non-trivial fact, namely that local homomorphisms between local Lie groups are automatically smooth; details are provided below the fold.

As with so many other basic classes of objects in mathematics, it is of fundamental importance to specify and study the morphisms between local groups (and group germs). Given two local groups {G, G'}, we can define the notion of a (continuous) homomorphism {\phi: G \rightarrow G'} between them, defined as a continuous map with

\displaystyle  \phi(1_G) = 1_{G'}

such that whenever {g, h \in G} are such that {gh} is well-defined, then {\phi(g)\phi(h)} is well-defined and equal to {\phi(gh)}; similarly, whenever {g \in G} is such that {g^{-1}} is well-defined, then {\phi(g)^{-1}} is well-defined and equal to {\phi(g^{-1})}. (In abstract algebra, the continuity requirement is omitted from the definition of a homomorphism; we will call such maps discrete homomorphisms to distinguish them from the continuous ones which will be the ones studied here.)

It is often more convenient to work locally: define a local (continuous) homomorphism {\phi: U \rightarrow G'} from {G} to {G'} to be a homomorphism from an open neighbourhood {U} of the identity to {G'}. Given two local homomorphisms {\phi: U \rightarrow G'}, {\tilde \phi: \tilde U \rightarrow \tilde G'} from one pair of locally identical groups {G, \tilde G} to another pair {G', \tilde G'}, we say that {\phi, \phi'} are locally identical if they agree on some open neighbourhood of the identity in {U \cap \tilde U'} (note that it does not matter here whether we require openness in {G}, in {\tilde G}, or both). An equivalence class {[\phi]} of local homomorphisms will be called a germ homomorphism (or morphism for short) from the group germ {[G]} to the group germ {[G']}.

Exercise 5 Show that the class of group germs, equipped with the germ homomorphisms, becomes a category. (Strictly speaking, because group germs are themselves classes rather than sets, the collection of all group germs is a second-order class rather than a class, but this set-theoretic technicality can be resolved in a number of ways (e.g. by restricting all global and local groups under consideration to some fixed “universe”) and should be ignored for this exercise.)

As is usual in category theory, once we have a notion of a morphism, we have a notion of an isomorphism: two group germs {[G], [G']} are isomorphic if there are germ homomorphisms {\phi: [G] \rightarrow [G']}, {\psi: [G'] \rightarrow [G]} that invert each other. Lifting back to local groups, the associated notion is that of local isomorphism: two local groups {G, G'} are locally isomorphic if there exist local isomorphisms {\phi: U \rightarrow G'} and {\psi: U' \rightarrow G} from {G} to {G'} and from {G'} to {G} that locally invert each other, thus {\psi(\phi(g))=g} for {g \in G} sufficiently close to {1_G}, and {\phi(\psi(g))} for {g' \in G'} sufficiently close to {1_{G'}}. Note that all local properties of (global or local) groups that can be defined purely in terms of the group and topological structures will be preserved under local isomorphism. Thus, for instance, if {G, G'} are locally isomorphic local groups, then {G} is locally connected iff {G'} is, {G} is locally compact iff {G'} is, and (by Lemma 1) {G} is Lie iff {G'} is.

Exercise 6

  • Show that the additive global groups {{\bf R}/{\bf Z}} and {{\bf R}} are locally isomorphic.
  • Show that every locally path-connected group {G} is locally isomorphic to a path-connected, simply connected group.
  • — 1. Lie’s third theorem —

    Lie’s fundamental theorems of Lie theory link the Lie group germs to Lie algebras. Observe that if {[G]} is a locally Lie group germ, then the tangent space {{\mathfrak g} := T_1 G} at the identity of this germ is well-defined, and is a finite-dimensional vector space. If we choose {G} to be symmetric, then {{\mathfrak g}} can also be identified with the left-invariant (say) vector fields on {G}, which are first-order differential operators on {C^\infty(M)}. The Lie bracket for vector fields then endows {{\mathfrak g}} with the structure of a Lie algebra. It is easy to check that every morphism {\phi: [G] \rightarrow [H]} of locally Lie germs gives rise (via the derivative map at the identity) to a morphism {D\phi(1): {\mathfrak g} \rightarrow {\mathfrak h}} of the associated Lie algebras. From the Baker-Campbell-Hausdorff formula (which is valid for local Lie groups, as discussed in this previous post) we conversely see that {D\phi(1)} uniquely determines the germ homomorphism {\phi}. Thus the derivative map provides a covariant functor from the category of locally Lie group germs to the category of (finite-dimensional) Lie algebras. In fact, this functor is an isomorphism, which is part of a fact known as Lie’s third theorem:

    Theorem 2 (Lie’s third theorem) For this theorem, all Lie algebras are understood to be finite dimensional (and over the reals).

    1. Every Lie algebra {{\mathfrak g}} is the Lie algebra of a local Lie group germ {[G]}, which is unique up to germ isomorphism (fixing {{\mathfrak g}}).
    2. Every Lie algebra {{\mathfrak g}} is the Lie algebra of some global connected, simply connected Lie group {G}, which is unique up to Lie group isomorphism (fixing {{\mathfrak g}}).
    3. Every homomorphism {\Phi: {\mathfrak g} \rightarrow {\mathfrak h}} between Lie algebras is the derivative of a unique germ homomorphism {\phi: [G] \rightarrow [H]} between the associated local Lie group germs.
    4. Every homomorphism {\Phi: {\mathfrak g} \rightarrow {\mathfrak h}} between Lie algebras is the derivative of a unique Lie group homomorphism {\phi: G \rightarrow H} between the associated global connected, simply connected, Lie groups.
    5. Every local Lie group germ is the germ of a global connected, simply connected Lie group {G}, which is unique up to Lie group isomorphism. In particular, every local Lie group is locally isomorphic to a global Lie group.

    We record the (standard) proof of this theorem below the fold, which is ultimately based on Ado’s theorem and the Baker-Campbell-Hausdorff formula. Lie’s third theorem (which, actually, was proven in full generality by Cartan) demonstrates the equivalence of three categories: the category of finite-dimensonal Lie algebras, the category of local Lie group germs, and the category of connected, simply connected Lie groups.

    — 2. Globalising a local group —

    Many properties of a local group improve after passing to a smaller neighbourhood of the identity. Here are some simple examples:

    Exercise 7 Let {G} be a local group.

    • Give an example to show that {G} does not necessarily obey the cancellation laws

      \displaystyle  gk=hk \implies g=h; \quad kg=kh \implies g=h \ \ \ \ \ (1)

      for {g,h,k \in G} (with the convention that statements such as {gk=hk} are false if either side is undefined). However, show that there exists an open neighbourhood {U} of {G} within which the cancellation law holds.

    • Repeat the previous part, but with the cancellation law (1) replaced by the inversion law

      \displaystyle  (gh)^{-1} = h^{-1} g^{-1} \ \ \ \ \ (2)

      for any {g,h \in G} for which both sides are well-defined.

    • Repeat the previous part, but with the inversion law replaced by the involution law

      \displaystyle  (g^{-1})^{-1} = g \ \ \ \ \ (3)

      for any {g} for which the left-hand side is well-defined.

    Note that the counterexamples in the above exercise demonstrate that not every local group is the restriction of a global group, because global groups (and hence, their restrictions) always obey the cancellation law (1), the inversion law (2), and the involution law (3). Another way in which a local group can fail to come from a global group is if it contains relations which can interact in a “global’ way to cause trouble, in a fashion which is invisible at the local level. For instance, consider the open unit cube {(-1,1)^3}, and consider four points {a_1, a_2, a_3, a_4} in this cube that are close to the upper four corners {(1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1)} of this cube respectively. Define an equivalence relation {\sim} on this cube by setting {x \sim y} if {x, y \in (-1,1)^3} and {x-y} is equal to either {0} or {\pm 2a_i} for some {i=1,\ldots,4}. Note that this indeed an equivalence relation if {a_1,a_2,a_3,a_4} are close enough to the corners (as this forces all non-trivial combinations {\pm 2a_i \pm 2a_j} to lie outside the doubled cube {(-2,2)^3}). The quotient space {(-1,1)^3/\sim} (which is a cube with bits around opposite corners identified together) can then be seen to be a symmetric additive local Lie group, but will usually not come from a global group. Indeed, it is not hard to see that if {(-1,1)^3/\sim} is the restriction of a global group {G}, then {G} must be a Lie group with Lie algebra {{\bf R}^3} (by Lemma 1), and so the connected component {G^\circ} of {G} containing the identity is isomorphic to {{\bf R}^3/\Gamma} for some sublattice {\Gamma} of {{\bf R}^3} that contains {a_1,a_2,a_3,a_4}; but for generic {a_1,a_2,a_3,a_4}, there is no such lattice, as the {a_i} will generate a dense subset of {{\bf R}^3}. (The situation here is somewhat analogous to a number of famous Escher prints, such as Ascending and Descending, in which the geometry is locally consistent but globally inconsistent.) We will give this sort of argument in more detail below the fold (see the proof of Proposition 7).

    Nevertheless, the space {(-1,1)^3/\sim} is still locally isomorphic to a global Lie group, namely {{\bf R}^3}; for instance, the open neighbourhood {(-0.5,0.5)^3/\sim} is isomorphic to {(-0.5,0.5)^3}, which is an open neighbourhood of {{\bf R}^3}. More generally, Lie’s third theorem tells us that any local Lie group is locally isomorphic to a global Lie group.

    Let us call a local group globalisable if it is locally isomorphic to a global group; thus Lie’s third theorem tells us that every local Lie group is globalisable. Thanks to Goldbring’s solution to the local version of Hilbert’s fifth problem, we also know that locally Euclidean local groups are globalisable. A modification of this argument by van den Dries and Goldbring shows in fact that every locally compact local group is globalisable.

    In view of these results, it is tempting to conjecture that all local groups are globalisable;; among other things, this would simplify the proof of Lie’s third theorem (and of the local version of Hilbert’s fifth problem). Unfortunately, this claim as stated is false:

    Theorem 3 There exists local groups {G} which are not globalisable.

    The counterexamples used to establish Theorem 3 are remarkably delicate; the first example I know of is due to van Est and Korthagen. One reason for this, of course, is that the previous results prevents one from using any local Lie group, or even a locally compact group as a counterexample. We will present a (somewhat complicated) example below, based on the unit ball in the infinite-dimensional Banach space {\ell^\infty({\bf N}^2)}.

    However, there are certainly many situations in which we can globalise a local group. For instance, this is the case if one has a locally faithful representation of that local group inside a global group:

    Lemma 4 (Faithful representation implies globalisability) Let {G} be a local group, and suppose there exists an injective local homomorphism {\phi: U \rightarrow H} from {G} into a global topological group {H} with {U} symmetric. Then {U} is isomorphic to the restriction of a global topological group to an open neighbourhood of the identity; in particular, {G} is globalisable.

    The material here is based in part on this paper of Olver and this paper of Goldbring.

    Read the rest of this entry »

    Over the past few months or so, I have been brushing up on my Lie group theory, as part of my project to fully understand the theory surrounding Hilbert’s fifth problem. Every so often, I encounter a basic fact in Lie theory which requires a slightly non-trivial “trick” to prove; I am recording two of them here, so that I can find these tricks again when I need to.

    The first fact concerns the exponential map {\exp: {\mathfrak g} \rightarrow G} from a Lie algebra {{\mathfrak g}} of a Lie group {G} to that group. (For this discuss we will only consider finite-dimensional Lie groups and Lie algebras over the reals {{\bf R}}.) A basic fact in the subject is that the exponential map is locally a homeomorphism: there is a neighbourhood of the origin in {{\mathfrak g}} that is mapped homeomorphically by the exponential map to a neighbourhood of the identity in {G}. This local homeomorphism property is the foundation of an important dictionary between Lie groups and Lie algebras.

    It is natural to ask whether the exponential map is globally a homeomorphism, and not just locally: in particular, whether the exponential map remains both injective and surjective. For instance, this is the case for connected, simply connected, nilpotent Lie groups (as can be seen from the Baker-Campbell-Hausdorff formula.)

    The circle group {S^1}, which has {{\bf R}} as its Lie algebra, already shows that global injectivity fails for any group that contains a circle subgroup, which is a huge class of examples (including, for instance, the positive dimensional compact Lie groups, or non-simply-connected Lie groups). Surjectivity also obviously fails for disconnected groups, since the Lie algebra is necessarily connected, and so the image under the exponential map must be connected also. However, even for connected Lie groups, surjectivity can fail. To see this, first observe that if the exponential map was surjective, then every group element {g \in G} has a square root (i.e. an element {h \in G} with {h^2 = g}), since {\exp(x)} has {\exp(x/2)} as a square root for any {x \in {\mathfrak g}}. However, there exist elements in connected Lie groups without square roots. A simple example is provided by the matrix

    \displaystyle  g = \begin{pmatrix} -4 & 0 \\ 0 & -1/4 \end{pmatrix}

    in the connected Lie group {SL_2({\bf R})}. This matrix has eigenvalues {-4}, {-1/4}. Thus, if {h \in SL_2({\bf R})} is a square root of {g}, we see (from the Jordan normal form) that it must have at least one eigenvalue in {\{-2i,+2i\}}, and at least one eigenvalue in {\{-i/2,i/2\}}. On the other hand, as {h} has real coefficients, the complex eigenvalues must come in conjugate pairs {\{ a+bi, a-bi\}}. Since {h} can only have at most {2} eigenvalues, we obtain a contradiction.

    However, there is an important case where surjectivity is recovered:

    Proposition 1 If {G} is a compact connected Lie group, then the exponential map is surjective.

    Proof: The idea here is to relate the exponential map in Lie theory to the exponential map in Riemannian geometry. We first observe that every compact Lie group {G} can be given the structure of a Riemannian manifold with a bi-invariant metric. This can be seen in one of two ways. Firstly, one can put an arbitrary positive definite inner product on {{\mathfrak g}} and average it against the adjoint action of {G} using Haar probability measure (which is available since {G} is compact); this gives an ad-invariant positive-definite inner product on {{\mathfrak g}} that one can then translate by either left or right translation to give a bi-invariant Riemannian structure on {G}. Alternatively, one can use the Peter-Weyl theorem to embed {G} in a unitary group {U(n)}, at which point one can induce a bi-invariant metric on {G} from the one on the space {M_n({\bf C}) \equiv {\bf C}^{n^2}} of {n \times n} complex matrices.

    As {G} is connected and compact and thus complete, we can apply the Hopf-Rinow theorem and conclude that any two points are connected by at least one geodesic, so that the Riemannian exponential map from {{\mathfrak g}} to {G} formed by following geodesics from the origin is surjective. But one can check that the Lie exponential map and Riemannian exponential map agree; for instance, this can be seen by noting that the group structure naturally defines a connection on the tangent bundle which is both torsion-free and preserves the bi-invariant metric, and must therefore agree with the Levi-Civita metric. (Alternatively, one can embed into a unitary group {U(n)} and observe that {G} is totally geodesic inside {U(n)}, because the geodesics in {U(n)} can be described explicitly in terms of one-parameter subgroups.) The claim follows. \Box

    Remark 1 While it is quite nice to see Riemannian geometry come in to prove this proposition, I am curious to know if there is any other proof of surjectivity for compact connected Lie groups that does not require explicit introduction of Riemannian geometry concepts.

    The other basic fact I learned recently concerns the algebraic nature of Lie groups and Lie algebras. An important family of examples of Lie groups are the algebraic groups – algebraic varieties with a group law given by algebraic maps. Given that one can always automatically upgrade the smooth structure on a Lie group to analytic structure (by using the Baker-Campbell-Hausdorff formula), it is natural to ask whether one can upgrade the structure further to an algebraic structure. Unfortunately, this is not always the case. A prototypical example of this is given by the one-parameter subgroup

    \displaystyle  G := \{ \begin{pmatrix} t & 0 \\ 0 & t^\alpha \end{pmatrix}: t \in {\bf R}^+ \} \ \ \ \ \ (1)

    of {GL_2({\bf R})}. This is a Lie group for any exponent {\alpha \in {\bf R}}, but if {\alpha} is irrational, then the curve that {G} traces out is not an algebraic subset of {GL_2({\bf R})} (as one can see by playing around with Puiseux series).

    This is not a true counterexample to the claim that every Lie group can be given the structure of an algebraic group, because one can give {G} a different algebraic structure than one inherited from the ambient group {GL_2({\bf R})}. Indeed, {G} is clearly isomorphic to the additive group {{\bf R}}, which is of course an algebraic group. However, a modification of the above construction works:

    Proposition 2 There exists a Lie group {G} that cannot be given the structure of an algebraic group.

    Proof: We use an example from the text of Tauvel and Yu (that I found via this MathOverflow posting). We consider the subgroup

    \displaystyle  G := \{ \begin{pmatrix} 1 & 0 & 0 \\ x & t & 0 \\ y & 0 & t^\alpha \end{pmatrix}: x, y \in {\bf R}; t \in {\bf R}^+ \}

    of {GL_3({\bf R})}, with {\alpha} an irrational number. This is a three-dimensional (metabelian) Lie group, whose Lie algebra {{\mathfrak g} \subset {\mathfrak gl}_3({\bf R})} is spanned by the elements

    \displaystyle  X := \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \alpha \end{pmatrix}

    \displaystyle  Y := \begin{pmatrix} 0 & 0 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

    \displaystyle  Z := \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ -\alpha & 0 & 0 \end{pmatrix}

    with the Lie bracket given by

    \displaystyle  [Y,X] = -Y; [Z,X] = -\alpha Z; [Y,Z] = 0.

    As such, we see that if we use the basis {X, Y, Z} to identify {{\mathfrak g}} to {{\bf R}^3}, then adjoint representation of {G} is the identity map.

    If {G} is an algebraic group, it is easy to see that the adjoint representation {\hbox{Ad}: G \rightarrow GL({\mathfrak g})} is also algebraic, and so {\hbox{Ad}(G) = G} is algebraic in {GL({\mathfrak g})}. Specialising to our specific example, in which adjoint representation is the identity, we conclude that if {G} has any algebraic structure, then it must also be an algebraic subgroup of {GL_3({\bf R})}; but {G} projects to the group (1) which is not algebraic, a contradiction. \Box

    A slight modification of the same argument also shows that not every Lie algebra is algebraic, in the sense that it is isomorphic to a Lie algebra of an algebraic group. (However, there are important classes of Lie algebras that are automatically algebraic, such as nilpotent or semisimple Lie algebras.)

    Hilbert’s fifth problem asks to clarify the extent that the assumption on a differentiable or smooth structure is actually needed in the theory of Lie groups and their actions. While this question is not precisely formulated and is thus open to some interpretation, the following result of Gleason and Montgomery-Zippin answers at least one aspect of this question:

    Theorem 1 (Hilbert’s fifth problem) Let {G} be a topological group which is locally Euclidean (i.e. it is a topological manifold). Then {G} is isomorphic to a Lie group.

    Theorem 1 can be viewed as an application of the more general structural theory of locally compact groups. In particular, Theorem 1 can be deduced from the following structural theorem of Gleason and Yamabe:

    Theorem 2 (Gleason-Yamabe theorem) Let {G} be a locally compact group, and let {U} be an open neighbourhood of the identity in {G}. Then there exists an open subgroup {G'} of {G}, and a compact subgroup {N} of {G'} contained in {U}, such that {G'/N} is isomorphic to a Lie group.

    The deduction of Theorem 1 from Theorem 2 proceeds using the Brouwer invariance of domain theorem and is discussed in this previous post. In this post, I would like to discuss the proof of Theorem 2. We can split this proof into three parts, by introducing two additional concepts. The first is the property of having no small subgroups:

    Definition 3 (NSS) A topological group {G} is said to have no small subgroups, or is NSS for short, if there is an open neighbourhood {U} of the identity in {G} that contains no subgroups of {G} other than the trivial subgroup {\{ \hbox{id}\}}.

    An equivalent definition of an NSS group is one which has an open neighbourhood {U} of the identity that every non-identity element {g \in G \backslash \{\hbox{id}\}} escapes in finite time, in the sense that {g^n \not \in U} for some positive integer {n}. It is easy to see that all Lie groups are NSS; we shall shortly see that the converse statement (in the locally compact case) is also true, though significantly harder to prove.

    Another useful property is that of having what I will call a Gleason metric:

    Definition 4 Let {G} be a topological group. A Gleason metric on {G} is a left-invariant metric {d: G \times G \rightarrow {\bf R}^+} which generates the topology on {G} and obeys the following properties for some constant {C>0}, writing {\|g\|} for {d(g,\hbox{id})}:

    • (Escape property) If {g \in G} and {n \geq 1} is such that {n \|g\| \leq \frac{1}{C}}, then {\|g^n\| \geq \frac{1}{C} n \|g\|}.
    • (Commutator estimate) If {g, h \in G} are such that {\|g\|, \|h\| \leq \frac{1}{C}}, then

      \displaystyle  \|[g,h]\| \leq C \|g\| \|h\|, \ \ \ \ \ (1)

      where {[g,h] := g^{-1}h^{-1}gh} is the commutator of {g} and {h}.

    For instance, the unitary group {U(n)} with the operator norm metric {d(g,h) := \|g-h\|_{op}} can easily verified to be a Gleason metric, with the commutator estimate (1) coming from the inequality

    \displaystyle  \| [g,h] - 1 \|_{op} = \| gh - hg \|_{op}

    \displaystyle  = \| (g-1) (h-1) - (h-1) (g-1) \|_{op}

    \displaystyle  \leq 2 \|g-1\|_{op} \|g-1\|_{op}.

    Similarly, any left-invariant Riemannian metric on a (connected) Lie group can be verified to be a Gleason metric. From the escape property one easily sees that all groups with Gleason metrics are NSS; again, we shall see that there is a partial converse.

    Remark 1 The escape and commutator properties are meant to capture “Euclidean-like” structure of the group. Other metrics, such as Carnot-Carathéodory metrics on Carnot Lie groups such as the Heisenberg group, usually fail one or both of these properties.

    The proof of Theorem 2 can then be split into three subtheorems:

    Theorem 5 (Reduction to the NSS case) Let {G} be a locally compact group, and let {U} be an open neighbourhood of the identity in {G}. Then there exists an open subgroup {G'} of {G}, and a compact subgroup {N} of {G'} contained in {U}, such that {G'/N} is NSS, locally compact, and metrisable.

    Theorem 6 (Gleason’s lemma) Let {G} be a locally compact metrisable NSS group. Then {G} has a Gleason metric.

    Theorem 7 (Building a Lie structure) Let {G} be a locally compact group with a Gleason metric. Then {G} is isomorphic to a Lie group.

    Clearly, by combining Theorem 5, Theorem 6, and Theorem 7 one obtains Theorem 2 (and hence Theorem 1).

    Theorem 5 and Theorem 6 proceed by some elementary combinatorial analysis, together with the use of Haar measure (to build convolutions, and thence to build “smooth” bump functions with which to create a metric, in a variant of the analysis used to prove the Birkhoff-Kakutani theorem); Theorem 5 also requires Peter-Weyl theorem (to dispose of certain compact subgroups that arise en route to the reduction to the NSS case), which was discussed previously on this blog.

    In this post I would like to detail the final component to the proof of Theorem 2, namely Theorem 7. (I plan to discuss the other two steps, Theorem 5 and Theorem 6, in a separate post.) The strategy is similar to that used to prove von Neumann’s theorem, as discussed in this previous post (and von Neumann’s theorem is also used in the proof), but with the Gleason metric serving as a substitute for the faithful linear representation. Namely, one first gives the space {L(G)} of one-parameter subgroups of {G} enough of a structure that it can serve as a proxy for the “Lie algebra” of {G}; specifically, it needs to be a vector space, and the “exponential map” needs to cover an open neighbourhood of the identity. This is enough to set up an “adjoint” representation of {G}, whose image is a Lie group by von Neumann’s theorem; the kernel is essentially the centre of {G}, which is abelian and can also be shown to be a Lie group by a similar analysis. To finish the job one needs to use arguments of Kuranishi and of Gleason, as discussed in this previous post.

    The arguments here can be phrased either in the standard analysis setting (using sequences, and passing to subsequences often) or in the nonstandard analysis setting (selecting an ultrafilter, and then working with infinitesimals). In my view, the two approaches have roughly the same level of complexity in this case, and I have elected for the standard analysis approach.

    Remark 2 From Theorem 7 we see that a Gleason metric structure is a good enough substitute for smooth structure that it can actually be used to reconstruct the entire smooth structure; roughly speaking, the commutator estimate (1) allows for enough “Taylor expansion” of expressions such as {g^n h^n} that one can simulate the fundamentals of Lie theory (in particular, construction of the Lie algebra and the exponential map, and its basic properties. The advantage of working with a Gleason metric rather than a smoother structure, though, is that it is relatively undemanding with regards to regularity; in particular, the commutator estimate (1) is roughly comparable to the imposition {C^{1,1}} structure on the group {G}, as this is the minimal regularity to get the type of Taylor approximation (with quadratic errors) that would be needed to obtain a bound of the form (1). We will return to this point in a later post.

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    Recall that a (complex) abstract Lie algebra is a complex vector space {{\mathfrak g}} (either finite or infinite dimensional) equipped with a bilinear antisymmetric form {[]: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}} that obeys the Jacobi identity

    \displaystyle  [[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0. \ \ \ \ \ (1)

    (One can of course define Lie algebras over other fields than the complex numbers {{\bf C}}, but in order to avoid some technical issues we shall work solely with the complex case in this post.)

    An important special case of the abstract Lie algebras are the concrete Lie algebras, in which {{\mathfrak g} \subset \hbox{End}(V)} is a vector space of linear transformations {X: V \rightarrow V} on a vector space {V} (which again can be either finite or infinite dimensional), and the bilinear form is given by the usual Lie bracket

    \displaystyle  [X,Y] := XY-YX.

    It is easy to verify that every concrete Lie algebra is an abstract Lie algebra. In the converse direction, we have

    Theorem 1 Every abstract Lie algebra is isomorphic to a concrete Lie algebra.

    To prove this theorem, we introduce the useful algebraic tool of the universal enveloping algebra {U({\mathfrak g})} of the abstract Lie algebra {{\mathfrak g}}. This is the free (associative, complex) algebra generated by {{\mathfrak g}} (viewed as a complex vector space), subject to the constraints

    \displaystyle  [X,Y] = XY - YX. \ \ \ \ \ (2)

    This algebra is described by the Poincaré-Birkhoff-Witt theorem, which asserts that given an ordered basis {(X_i)_{i \in I}} of {{\mathfrak g}} as a vector space, that a basis of {U({\mathfrak g})} is given by “monomials” of the form

    \displaystyle  X_{i_1}^{a_1} \ldots X_{i_m}^{a_m} \ \ \ \ \ (3)

    where {m} is a natural number, the {i_1 < \ldots < i_m} are an increasing sequence of indices in {I}, and the {a_1,\ldots,a_m} are positive integers. Indeed, given two such monomials, one can express their product as a finite linear combination of further monomials of the form (3) after repeatedly applying (2) (which we rewrite as {XY = YX + [X,Y]}) to reorder the terms in this product modulo lower order terms until one all monomials have their indices in the required increasing order. It is then a routine exercise in basic abstract algebra (using all the axioms of an abstract Lie algebra) to verify that this is multiplication rule on monomials does indeed define a complex associative algebra which has the universal properties required of the universal enveloping algebra.

    The abstract Lie algebra {{\mathfrak g}} acts on its universal enveloping algebra {U({\mathfrak g})} by left-multiplication: {X: M \mapsto XM}, thus giving a map from {{\mathfrak g}} to {\hbox{End}(U({\mathfrak g}))}. It is easy to verify that this map is a Lie algebra homomorphism (so this is indeed an action (or representation) of the Lie algebra), and this action is clearly faithful (i.e. the map from {{\mathfrak g}} to {\hbox{End}(U{\mathfrak g})} is injective), since each element {X} of {{\mathfrak g}} maps the identity element {1} of {U({\mathfrak g})} to a different element of {U({\mathfrak g})}, namely {X}. Thus {{\mathfrak g}} is isomorphic to its image in {\hbox{End}(U({\mathfrak g}))}, proving Theorem 1.

    In the converse direction, every representation {\rho: {\mathfrak g} \rightarrow \hbox{End}(V)} of a Lie algebra “factors through” the universal enveloping algebra, in that it extends to an algebra homomorphism from {U({\mathfrak g})} to {\hbox{End}(V)}, which by abuse of notation we shall also call {\rho}.

    One drawback of Theorem 1 is that the space {U({\mathfrak g})} that the concrete Lie algebra acts on will almost always be infinite-dimensional, even when the original Lie algebra {{\mathfrak g}} is finite-dimensional. However, there is a useful theorem of Ado that rectifies this:

    Theorem 2 (Ado’s theorem) Every finite-dimensional abstract Lie algebra is isomorphic to a concrete Lie algebra over a finite-dimensional vector space {V}.

    Among other things, this theorem can be used (in conjunction with the Baker-Campbell-Hausdorff formula) to show that every abstract (finite-dimensional) Lie group (or abstract local Lie group) is locally isomorphic to a linear group. (It is well-known, though, that abstract Lie groups are not necessarily globally isomorphic to a linear group, but we will not discuss these global obstructions here.)

    Ado’s theorem is surprisingly tricky to prove in general, but some special cases are easy. For instance, one can try using the adjoint representation {\hbox{ad}: {\mathfrak g} \rightarrow \hbox{End}({\mathfrak g})} of {{\mathfrak g}} on itself, defined by the action {X: Y \mapsto [X,Y]}; the Jacobi identity (1) ensures that this indeed a representation of {{\mathfrak g}}. The kernel of this representation is the centre {Z({\mathfrak g}) := \{ X \in {\mathfrak g}: [X,Y]=0 \hbox{ for all } Y \in {\mathfrak g}\}}. This already gives Ado’s theorem in the case when {{\mathfrak g}} is semisimple, in which case the center is trivial.

    The adjoint representation does not suffice, by itself, to prove Ado’s theorem in the non-semisimple case. However, it does provide an important reduction in the proof, namely it reduces matters to showing that every finite-dimensional Lie algebra {{\mathfrak g}} has a finite-dimensional representation {\rho: {\mathfrak g} \rightarrow \hbox{End}(V)} which is faithful on the centre {Z({\mathfrak g})}. Indeed, if one has such a representation, one can then take the direct sum of that representation with the adjoint representation to obtain a new finite-dimensional representation which is now faithful on all of {{\mathfrak g}}, which then gives Ado’s theorem for {{\mathfrak g}}.

    It remins to find a finite-dimensional representation of {{\mathfrak g}} which is faithful on the centre {Z({\mathfrak g})}. In the case when {{\mathfrak g}} is abelian, so that the centre {Z({\mathfrak g})} is all of {{\mathfrak g}}, this is again easy, because {{\mathfrak g}} then acts faithfully on {{\mathfrak g} \times {\bf C}} by the infinitesimal shear maps {X: (Y,t) \mapsto (tX, 0)}. In matrix form, this representation identifies each {X} in this abelian Lie algebra with an “upper-triangular” matrix:

    \displaystyle  X \equiv \begin{pmatrix} 0 & X \\ 0 & 0 \end{pmatrix}.

    This construction gives a faithful finite-dimensional representation of the centre {Z({\mathfrak g})} of any finite-dimensional Lie algebra. The standard proof of Ado’s theorem (which I believe dates back to work of Harish-Chandra) then proceeds by gradually “extending” this representation of the centre {Z({\mathfrak g})} to larger and larger sub-algebras of {{\mathfrak g}}, while preserving the finite-dimensionality of the representation and the faithfulness on {Z({\mathfrak g})}, until one obtains a representation on the entire Lie algebra {{\mathfrak g}} with the required properties. (For technical inductive reasons, one also needs to carry along an additional property of the representation, namely that it maps the nilradical to nilpotent elements, but we will discuss this technicality later.)

    This procedure is a little tricky to execute in general, but becomes simpler in the nilpotent case, in which the lower central series {{\mathfrak g}_1 := {\mathfrak g}; {\mathfrak g}_{n+1} := [{\mathfrak g}, {\mathfrak g}_n]} becomes trivial for sufficiently large {n}:

    Theorem 3 (Ado’s theorem for nilpotent Lie algebras) Let {{\mathfrak n}} be a finite-dimensional nilpotent Lie algebra. Then there exists a finite-dimensional faithful representation {\rho: {\mathfrak n} \rightarrow \hbox{End}(V)} of {{\mathfrak n}}. Furthermore, there exists a natural number {k} such that {\rho({\mathfrak n})^k = \{0\}}, i.e. one has {\rho(X_1) \ldots \rho(X_k)=0} for all {X_1,\ldots,X_k \in {\mathfrak n}}.

    The second conclusion of Ado’s theorem here is useful for induction purposes. (By Engel’s theorem, this conclusion is also equivalent to the assertion that every element of {\rho({\mathfrak n})} is nilpotent, but we can prove Theorem 3 without explicitly invoking Engel’s theorem.)

    Below the fold, I give a proof of Theorem 3, and then extend the argument to cover the full strength of Ado’s theorem. This is not a new argument – indeed, I am basing this particular presentation from the one in Fulton and Harris – but it was an instructive exercise for me to try to extract the proof of Ado’s theorem from the more general structural theory of Lie algebras (e.g. Engel’s theorem, Lie’s theorem, Levi decomposition, etc.) in which the result is usually placed. (However, the proof I know of still needs Engel’s theorem to establish the solvable case, and the Levi decomposition to then establish the general case.)

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    In this final lecture, we establish a Ratner-type theorem for actions of the special linear group SL_2({\Bbb R}) on homogeneous spaces. More precisely, we show:

    Theorem 1. Let G be a Lie group, let \Gamma < G be a discrete subgroup, and let H \leq G be a subgroup isomorphic to SL_2({\Bbb R}). Let \mu be an H-invariant probability measure on G/\Gamma which is ergodic with respect to H (i.e. all H-invariant sets either have full measure or zero measure). Then \mu is homogeneous in the sense that there exists a closed connected subgroup H \leq L \leq G and a closed orbit Lx \subset G/\Gamma such that \mu is L-invariant and supported on Lx.

    This result is a special case of a more general theorem of Ratner, which addresses the case when H is generated by elements which act unipotently on the Lie algebra {\mathfrak g} by conjugation, and when G/\Gamma has finite volume. To prove this theorem we shall follow an argument of Einsiedler, which uses many of the same ingredients used in Ratner’s arguments but in a simplified setting (in particular, taking advantage of the fact that H is semisimple with no non-trivial compact factors). These arguments have since been extended and made quantitative by Einsiedler, Margulis, and Venkatesh.
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