Invariant complement from an equivariant projection
= Invariant complement from an equivariant projection
{title2=$V=U\oplus\ker P$}
If a submodule $U$ of a <Lie algebra representation> $V$ admits an intertwining map $P:V\to U$ with $P|_U=I_U$, then $P^2=P$ and its <kernel> is an invariant complement. For a complex <semisimple Lie algebra>, the <Hom representation> on maps $V\to U$ whose restriction is a scalar identity reduces finding $P$ to <splitting of a trivial quotient for a semisimple Lie algebra>.