Exercise 7, § 2

Lie Groups and Lie Algebras, II, 183

Suppose that K is a field. Let l be a free Lie algebra with basic family $(x_i)_{i \in I}$ and let M be an l-module. (a) Show that $H^2(l, M) = \{0\}$ , cf. Chapter I, § 3, Exercise 12. (Use Corollary 2 to Proposition 1 and part (i) of the Exercise in question.) (b) Show that, for every family $(m_i)_{i \in I}$ of elements of M, there exists a cocycle $\phi : l \to M$ of degree 1 such that $\phi(x_i) = m_i$ and that this cocycle is unique. Deduce an exact sequence:

0 \to H^0(l, M) \to M \to M^l \to H^1(l, M) \to 0.

If I is finite of cardinal n and M is of finite rank over K, then

\operatorname{rg} H^1(l, M) - \operatorname{rg} H^0(l, M) = (n - 1) \operatorname{rg} M.

Solution

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