Exercise 5, § 1

Let E be a locally finite $\mathfrak{s}$-module (Exerc. 4). For all $m\geq 0$, denote by $L(m)$ the vector space of $\mathfrak{s}$-homomorphisms from $V(m)$ to E.

a) Define an isomorphism from E to $\bigoplus_mL_m\otimes V(m)$.

b) Denote by $\Phi_m$ the invariant bilinear form on $V(m)$ defined in no. 3, Remark 3. For all $m\in \mathbf{N}$, let $b_m$ be a bilinear form on $L_m$; let $b$ be the bilinear form on E which corresponds, via the isomorphism in a), to the direct sum of the forms $b_m\otimes \Phi_m$. Show that $b$ is invariant, and that every invariant bilinear form on E is obtained in this way in a unique manner. The form $b$ is symmetric (resp. alternating) if and only if the $b_m,m$ even, are symmetric (resp. alternating), and the $b_m,m$ odd, are alternating (resp. symmetric). The form $b$ is non-degenerate if and only if the $b_m$ are non-degenerate.

c) Assume that E is finite dimensional. Show that E is monogenic (as a $U(\mathfrak{s})$-module) if and only if dim $L_m\leq m+ 1$ for all $m\geq 0$.