Exercise 6, § 5

Let $\mathfrak{g}$ be a semi-simple subalgebra of $\mathfrak{g}\mathfrak{l}$(V), $\mathbf{T}(V) =\bigoplus_{n=0}^{\infty}\mathbf{T}^n(V)$ the

tensor algebra of $V,\mathbf{T}(V)^{\mathfrak{g}}$ the set of elements of $\mathbf{T}(V)$ invariant under $\mathfrak{g}$ (cf. Chap. III, App.) and $\overline{\mathfrak{g}}$ the set of $u\in \mathfrak{g}\mathfrak{l}(V)$ such that $u.x= 0$ for all $x\in \mathbf{T}(V)^{\mathfrak{g}}$. We are going to show that $\overline{\mathfrak{g}}=\mathfrak{g}$.

a) Show that the representation of $\mathfrak{g}$ on the dual $V^*$ of V is isomorphic to its representation on $\bigwedge^{p-1}V$, where $p=$ dim V (use the fact that $\mathfrak{g}$ is contained in $\mathfrak{s}\mathfrak{l}(V)$). Deduce that every element of $\mathbf{T}_{n,m}=\mathbf{T}^n(V)\otimes \mathbf{T}^m(V^*)$ invariant under $\mathfrak{g}$ is invariant under $\overline{\mathfrak{g}}$, and that $\overline{\mathfrak{g}}$ is algebraic (Exerc. 5).

b) Let W be a vector subspace of one of the $\mathbf{T}_{n,m}$. Assume that W is stable under $\mathfrak{g}$. Show that W is stable under $\overline{\mathfrak{g}}$ (if $e_1, . . . , e_r$ is a basis of W, remark that $e_1\wedge \cdots \wedge e_r$ is invariant under $\mathfrak{g}$, and hence under $\overline{\mathfrak{g}}$).

Deduce that $\mathfrak{g}$ is an ideal of $\overline{\mathfrak{g}}$, and that $\overline{\mathfrak{g}}/\mathfrak{g}$ is commutative (cf. proof of Prop. 4). We have $\overline{\mathfrak{g}}=\mathfrak{g}\times \mathfrak{c}$, where $\mathfrak{c}$ is the centre of $\overline{\mathfrak{g}}$.

c) Let R be the associative subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ generated by 1 and $\mathfrak{g}$. Show that $\mathfrak{c}$ is contained in the centre of R (remark that $\overline{\mathfrak{g}}$ is contained in the bicommutant of R, which is equal to R). Deduce that the elements of $\mathfrak{c}$ are semi-simple.

d) Let $x\in \mathfrak{c}$. Show that the replicas of $x$ belong to $\mathfrak{c}$ (Chap. I, §5, Exerc. 14). Show that Tr($sx$) $= 0$ for all $s\in \mathfrak{g}$; deduce that Tr($sx$) $= 0$ for all $s\in \overline{\mathfrak{g}}$, and hence that $x$ is nilpotent (loc. cit).

e) Show that $\mathfrak{c}= 0$ and $\overline{\mathfrak{g}}=\mathfrak{g}$ by combining c) and d).