From: hook@nas.nasa.gov (Ed Hook)
Subject: Re: compact lie group/algebra
Date: 7 Jul 2000 14:14:05 GMT
Newsgroups: sci.math
Summary: [missing]

In article <39642461.685B758C@submaths.hku.hk>,
        cwwongreborn <cwwong@submaths.hku.hk> writes:

|> i've read the following argument among others, which may be basic in lie
|> group/algebra theory:
 
|> if G is a compact lie group, then its algebra admits a positive
|> definitive (symmetric) invariant bilinear form.
 
|> it seems to be tr(A^tB) which playing this role. but i don't see why
|> compactness is assumed.

  To get an idea of what's going on, let G
  be a *finite* group and suppose that V
  is a vector space on which G acts. Pick
  any bilinear form Q: VxV --> R -- then,
  using Q, you can define an invariant
  bilinear form Q* on V by setting

   Q*(v,w) =     Sum   Q(gv,gw)/|G|
               g \in G

  If the original Q is symmetric, so is
  Q*. If the original Q is positive
  definite, so is Q*.

  More generally, if G is a _compact_ Lie
  group (in fact, just a compact *topological*
  group suffices), then you can carry out
  essentially this same construction, replacing
  the average over the group above by the Haar
  integral over G of Q(g^{-1}v,g^{-1}w). (The
  inverses are in there, because you really
  need an action of G on the vector space of
  bilinear forms -- then the "translation
  invariance" of the Haar integral allows you
  to prove that the Q* you define is, in fact,
  an invariant form.)

  The Haar integral can be defined more generally,
  but you aren't guaranteed that you'll be able
  to integrate Q(<whatever>) except in the compact
  case.

-- 
 Ed Hook                              |       Copula eam, se non posit
 Computer Sciences Corporation        |         acceptera jocularum.
 NAS, NASA Ames Research Center       |    All opinions herein expressed are
 Internet: hook@nas.nasa.gov          |              mine alone