# invariant form (Lie algebras)

Let $V$ be a representation of a Lie algebra $\mathfrak{g}$ over a field $k$. Then a bilinear form $B:V\times V\to k$ is invariant if

 $B(Xv,w)+B(v,Xw)=0.$

for all $X\in\mathfrak{g},v,w\in V$. This criterion seems a little odd, but in the context of Lie algebras, it makes sense. For example, the map $\tilde{B}:V\to V^{*}$ given by $v\mapsto B(\cdot,v)$ is equivariant if and only if $B$ is an invariant form.

Title invariant form (Lie algebras) InvariantFormLieAlgebras 2013-03-22 13:15:45 2013-03-22 13:15:45 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 17B15