Exercise 3, § 3

Let V be a finite-dimensional vector space over a perfect field K. (a) The identity representation of $gl(V)$ defines canonically representations of $gl(V)$ on $\bigotimes^p V$ and $\bigotimes^q V^*$ and hence a representation $u \mapsto u_q^p$ of $gl(V)$ on

Show that $u_q^p$ is semi-simple if $u$ is. (Reduce it, by extending the base field, to the case where $K$ is algebraically closed.) Show that $u_q^p$ is nilpotent if $u$ is. Deduce that, if $s$ and $n$ are the semi-simple and nilpotent components of $u$, $s_q^p$ and $n_q^p$ are the semi-simple and nilpotent components of $u_q^p$.

(b) Deduce from (a) that, if an element of $V_q^p$ is annihilated by $u_q^p$, it is annihilated by $s_q^p$ and $n_q^p$.

(c) Deduce from (b) and Exercise 2 that, if V has a not necessarily associative algebra structure and u is a derivation of V, then s and n are derivations of V.