Exercise 4, § 7

Let $\mathfrak{g}$ be a $k$-Lie algebra, $M$ a $\mathfrak{g}$-module. An extension of $\mathfrak{g}$ by $M$ is an exact sequence of $k$-modules

where $\mathfrak{h}$ is a $k$-Lie algebra, $p$ a homomorphism of Lie algebras, and where $i(\rho(H)m)=[H,i(m)]$ for $H\in\mathfrak{h},m\in M$. Two extensions

are said to be equivalent if there exists a homomorphism $u:\mathfrak{h}\to\mathfrak{h}'$ such that $p'\circ u=p,\ u\circ i=i'$. We denote by $Lie_x(\mathfrak{g},M)$ the set of equivalence classes of extensions of $\mathfrak{g}$ by $M$. We suppose that $\mathfrak{g}$ is a free $k$-module.

a) Let $(\mathscr{E}):0\to M\to\mathfrak{h}\xrightarrow{p}\mathfrak{g}\to0$ be an extension of $\mathfrak{g}$ by $M$; choose a $k$-linear section $s:\mathfrak{g}\to\mathfrak{h}$ of $p$. We define an element $f\in C^2(\mathfrak{g},M)$ (X, p. 196, exercise 28, b)) by putting

Prove that $f$ is a $2$-cocycle and that its class in $H^2(\mathfrak{g},M)$ does not depend on the choice of $s$; it is denoted by $\theta(\mathscr{E})$.

b) Prove that $\theta$ defines a bijection of $Lie_x(\mathfrak{g},M)$ onto $H^2(\mathfrak{g},M)$. Describe directly the group structure on $Lie_x(\mathfrak{g},M)$ obtained by transport by $\gamma$.

c) Let U be the enveloping algebra of g, $M_{(\pi)}$ the (U, U)-bimodule associated to M by means of the augmentation $\pi : U \to k$ (X, p. 196, exercise 27). Let $0 \to M_{(\pi)} \to B \xrightarrow{\rho} U \to 0$ be an extension of U by $M_{(\pi)}$ (exercise 1); denote by b the set of elements b of B such that $p(b) \in j(g)$. Prove that b is a $k$-Lie algebra, an extension of g by M. This construction defines a mapping $\lambda : Ex_k(U, M_{(\pi)}) \to Liex(g, M)$. Prove that the diagram

(where $\varphi^2$ is the isomorphism defined in X, p. 196, exercise 27) is commutative. Deduce that $\lambda$ is an isomorphism. \*