invariant forms on representations of compact groups
Let be a real Lie group. TFAE:
One faithful representation of has an invariant positive definite form.
: Let be the invariant form on a faithful representation . Let then representation gives an embedding , the group of automorphisms of preserving . Thus, is homeomorphic to a closed subgroup of . Since this group is compact, must be compact as well.
(Proof that is compact: By induction on . Let be an arbitrary vector. Then there is a map, evaluation on , from (this is topologically a sphere, since is isometric to with the standard norm). This is a a fiber bundle, and the fiber over any point is a copy of , which is compact by the inductive hypothesis. Any fiber bundle over a compact base with compact fiber has compact total space. Thus is compact).
: Let be an arbitrary representation of . Choose an arbitrary positive definite form on . Then define
Furthermore, is invariant, since
For representation of the maximal torus , there exists a representation of , with a -subrepresentation of . Also, since every conjugacy class of intersects any maximal torus, a representation of is faithful if and only if it restricts to a faithful representation of . Since any torus has a faithful representation, must have one as well.
Given that these criteria hold, let be a representation of , is positive definite real form, and a subrepresentation. Now consider
By the positive definiteness of , . By induction, is completely reducible.
Applying this to the adjoint representation of on , its Lie algebra, we find that in the direct sum of simple algebras , in the sense that has no proper nontrivial ideals, meaning that is simple in the usual sense or it is abelian. ∎
|Title||invariant forms on representations of compact groups|
|Date of creation||2013-03-22 13:23:40|
|Last modified on||2013-03-22 13:23:40|
|Last modified by||bwebste (988)|