= Lie-invariant bilinear form
{c}
{title2=$B(xv,w)+B(v,xw)=0$}
A <bilinear form> on a <Lie algebra representation> is invariant when it satisfies the displayed identity. Equivalently, $T_B(v)=B(v,-)$ is an <intertwining operator> from the representation to its <dual Lie algebra representation>. For a finite-dimensional <irreducible representation> over an <algebraically closed field>, any nonzero such form is <nondegenerate>, and the <Schur lemma> makes all invariant forms proportional. In <characteristic> different from two, transposing twice then proves that the form is a <symmetric bilinear form> or an <alternating bilinear form>.
Back to article page
