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 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() 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