Exponential maps of Lie groups: applications

We first recall a theorem proved in “Exponential maps on Lie groups”:

Ken’s Theorem: Let G be a Lie groups with a bi-invariant metric (e.g. if G is compact). Then the exponential map exp: \mathfrak g \to G is the same as the Riemannian exponential map

Exp_e : T_eG \to G at the identity e \in G.

In this note we assume that G is compact, and we wish to derive some consequences of the above theorem. The most important one is the Cartan’s theorem proved below. Some information concerning the fundamental group of G can also be obtained. One might consult the two previous blogs by KKK on Lie groups for the background information needed.

First we state a lemma.

Lemma 1: The exponential map \mathfrak g \to G is surjective.

The lemma is a direct consequence of Ken’s theorem, because any metric defined on a compact manifold is complete, and completeness is equivalent to the fact that

\forall p\in G, Exp_p :T_pG \to G is surjective.

Next we introduce the notion of a maximal torus. By definition, a maximal torus T in G is a maximal connected abelian subgroup of G. Here maximal means that if T’ contains T and T’ is connected and abelian, then T' = T. One can check that T must be closed (because the closure of any subgroup is still a subgroup), and  T must be a torus

T \cong \mathbb S ^1 \times \cdots \times \mathbb S^1

by compactness of G (the proof is omitted, it depends on some analysis on the exponential map).

Note that maximal tours are not unique: if T is a maximal torus, then xTx^{-1} is also a maximal torus. Cartan’s theorem states that all maximal torus are of this form:

Cartan’s theorem: Let T, T’ be two maximal torus. Then there is a g\in G such that T' = gTg^{-1}.

We first state a proposition.

Proposition 1: Let T be a maximal torus and g\in G. Then \exists x\in G such that g \in xTx^{-1}.

Proof of proposition 1: Let g\in G and h_0 \in T be a generator of T. Here a generator means that \overline{\{h_0^n : n\in \mathbb N\}} =T (why exists?). Using Lemma 1, there is X \in \mathfrak g and H_0 \in \mathfrak t such that e^X = g and e^{H_0} = h_0. Let x\in G be choosen such that \langle X, Ad_{x} H_0 \rangle is maximized.  

Let H = Ad_{x} H_0 (Then h = e^H is a generator of xTx^{-1}). For any Y \in \mathfrak g, the term \langle X, Ad_{e^{tY}} H \rangle has a maximum at t=0. Thus

0 = (\langle X, Ad_{e^{tY}} H \rangle)'_{t=0} = \langle X, [Y,H]\rangle = \langle Y, [H,X]\rangle .

The proof for the second and thrid equalities can be found in KKK’s “Lie groups with bi-invariant Riemannian metric”. As Y is arbitrary, [H,X]=0 and e^{tX}h =he^{tX}. Thus e^{tX}h^n = h^n e^{tX} for any n and so e^{tX} commutes with every elements in xTx^{-1}. Let W be the connected abelian group generated by e^{tX} and xTx^{-1}. As xTx^{-1} is maximal abelian, we have W = xTx^{-1} and g = e^X\in xTx^{-1}.

Proof of Cartan’s theorem: Let t’ be a generator of T’. Then t' \in xTx^{-1} for some x, by proposition 1. Then T' \subset xTx^{-1}.  As T’ is maximal abelian, T' = xTx^{-1} and the theorem is proved.

Indeed, Cartan’s theorem is quite important in representation theory of compact Lie group. As T is abelian, representation on T is well known (just Fourier analysis). One hopes to know all representations of G by looking at that of T, and Cartan’s theorem tells us that all maximal abelian connected subgroups T are the same.

To end this note, we give an application of Cartan’s theorem. Let T be a maximal torus and

i : T \to G

be the inclusion map.

Theorem 2: The induced map i_{*}:\pi_1 (T) \to \pi _1 (G) is surjective. Thus the fundamental group of G is abelian and finitely generated.

Proof of theorem 2: Let [\gamma] \in \pi_1(G). Then by compactness of G, \gamma can be choosen to be a geodesic (see Jost for a detailed discription, at least it is intuitively clear…).  By a left multiplication (which is an isometry as the metric is bi-invariant) if neccesary, we can assumed that this geodesic starts at the identity e in G. Then \gamma(t) = e^{tX} for some X \in \mathfrak g. Note that this subset forms an abelian subgroup of G, thus is contained in some maximal torus T’. By Cartan’s theorem, T' = xTx^{-1} for some x and

\gamma (t) = x \beta(t) x^{-1},

where \beta(t) \in T. Let x(u) be a path joining e to x. Then obviously

A(t,u) = x(u) \beta(t) x(u)^{-1}

forms a homotopy between \gamma and \beta. Thus i_* ([\beta]) = [\gamma] and the theorem is proved.

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2 Responses to Exponential maps of Lie groups: applications

  1. KKK says:

    To avoid confusion, in the proof of Proposition 1, \langle X, Ad_{e(tY)}H\rangle should be written as \langle X, Ad_{e^{tY}}H\rangle instead (there are two instances).

  2. Anonymous says:

    Changed as suggested, Thank you~

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