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 is the same as the Riemannian exponential map
at the identity .
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 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
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 . One can check that T must be closed (because the closure of any subgroup is still a subgroup), and T must be a torus
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 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 such that .
We first state a proposition.
Proposition 1: Let T be a maximal torus and . Then such that .
Proof of proposition 1: Let and be a generator of T. Here a generator means that (why exists?). Using Lemma 1, there is and such that and . Let be choosen such that is maximized.
Let (Then is a generator of ). For any , the term has a maximum at . Thus
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, and . Thus for any n and so commutes with every elements in . Let W be the connected abelian group generated by and . As is maximal abelian, we have and .
Proof of Cartan’s theorem: Let t’ be a generator of T’. Then for some x, by proposition 1. Then . As T’ is maximal abelian, 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
be the inclusion map.
Theorem 2: The induced map is surjective. Thus the fundamental group of G is abelian and finitely generated.
Proof of theorem 2: Let . Then by compactness of G, 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 for some . Note that this subset forms an abelian subgroup of G, thus is contained in some maximal torus T’. By Cartan’s theorem, for some x and
where . Let be a path joining e to x. Then obviously
forms a homotopy between and . Thus and the theorem is proved.