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 discussion 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.)