= Invariant form on an irreducible sl2 module
In the irreducible <sl2 Lie algebra> module $V(m)$, use $v_j=f^jv_0$, with $0\le j\le m$. The <Lie-invariant bilinear form> $B(v_j,v_k)=(-1)^j\mathbf1_{\{j+k=m\}}$ is <nondegenerate>: its anti-diagonal is nonzero. Invariance follows from $hv_j=(m-2j)v_j$, $fv_j=v_{j+1}$ and $ev_j=j(m-j+1)v_{j-1}$. Its transpose equals $(-1)^mB$, so it is a <symmetric bilinear form> when $m$ is an <even number>, and an <alternating bilinear form> when $m$ is an <odd integer>. The <Weyl complete reducibility theorem> gives a nondegenerate invariant form on any finite-dimensional module by taking the orthogonal <direct sum> of these forms.
Back to article page
