Exercise 12, § 3

Suppose that $K$ is a field. Let $g$ be a Lie $K$-algebra and $M$ a $g$-module. Let $C^p(g, M)$ denote the space of alternating linear mappings of $g^p$ into $M$. We write $C^0(g, M) = M$ and, for $p < 0$, $C^p(g, M) = \{0\}$. Let $C^*(g, M)$ be the direct sum of the $C^p(g, M)$. The elements of $C^*(g, M)$ are called the cochains of $g$ with values in $M$; those of $C^p(g, M)$ are said to be of degree $p$. For all $y \in g$, we denote by $i(y)$ the endomorphism of $C^*(g, M)$ which maps each subspace $C^p(g, M)$ into $C^{p-1}(g, M)$ and which, for $p > 0$, is given by the formula:

We have $i(y)^2 = 0$.

(a) The adjoint representation of $g$ and the representation of $g$ on $M$ define a representation of $g$ on the space of multilinear mappings of $g^p$ into $M$. Show that $C^p(g, M)$ is stable under this representation. Let $\theta$ be the representation of $g$ on $C^*(g, M)$ thus defined. Show that:

for all $x, y$ in $g$.

(b) Show that there exists one and only one endomorphism $d$ of $C^*(g, M)$, mapping $C^p(g, M)$ into $C^{p+1}(g, M)$, such that

for all $y \in g$. (Argue by induction on the degree of the cochains.) Show that, for $f \in C^p(g, M)$:

(where the symbol ^ over a letter means that it is omitted).

(c) Show that, for all $y \in g$,

(Show first, using (2) and (3), that $d \theta(y) - \theta(y)d$ commutes with every $i(x)$. Then argue by induction on the degree of the cochains.)

(d) Show that $d^2 = 0$. (Show first, using (3) and (4), that $d^2$ commutes with every $i(x)$. The argue by induction on the degree of the cochains.)

(e) The restriction of $d$ to $C^p(g, M)$ has a kernel $Z^p(g, M)$ whose elements are called cocycles of degree $p$ with values in $M$. The restriction of $d$ to $C^{p-1}(g, M)$ has image $B^p(g, M)$, whose elements are called coboundaries of degree $p$ with values in $M$. Then $B^p(g, M) \subset Z^p(g, M)$. The quotient space

is called the cohomology space of $g$ of degree $p$ with values in $M$. The direct sum of the $H^p(g, M)$ is denoted by $H^*(g, M)$. Show that $H^0(g, M)$ is identified with the set of invariants of $M$. Let $\phi$ be a homomorphism of the $g$-module $M$ into the $G$-module $N$. For all $f \in C^p(g, M)$, $\phi \circ f \in C^p(g, N)$, so that $\phi$ can be extended to a K-linear mapping $\phi': C^*(g, M) \to C^*(g, N)$. Show that $\phi' \circ d = d \circ \phi'$. Deduce that $\phi'$ defines a homomorphism

which is said to be associated with $\phi$.

(f) Let $L$ be a sub-$g$-module of $M$ and $N$ the quotient $g$-module $M/L$. The canonical homomorphisms $L \xrightarrow{i} M \xrightarrow{p} N$ define homomorphisms

Show that $i'$ is injective and has image the kernel of $p'$ and that $p'$ is surjective. For all $c \in H^n(g, N)$, let $z \in Z^n(g, N)$ be a representative of $c$ and $a \in C^n(g, M)$ be such that $p'(a) = z$; show that $da \in Z^{n+1}(g, L)$ and that its class in $H^{n+1}(g, L)$ depends only on $c$; denote this class by $\delta^n c$. Show that the sequence of homomorphisms:

is an exact sequence.

(g) With the notation of (f), the exact sequence:

defines an exact sequence

The identity mapping of N is an invariant u of $\mathscr{L}(N, N)$ and hence an element of $H^0(g, \mathscr{L}(N, N))$; let c be its image in $H^1(g, \mathscr{L}(N, L))$. Then for the existence in M of a sub-g-module supplementary to L it is necessary and sufficient that $c = 0$. (This condition means that u is the image of an element of $H^0(g, \mathscr{L}(N, M))$, that is a homomorphism v of the g-module N into the g-module M; then $v(N)$ is the desired supplement.)

(h) Show that $Z^1(g, g)$ (where g is considered as a g-module by means of the adjoint representation) is identified with the vector space of derivations of g and that $B^1(g, g)$ is identified with the vector space of inner derivations of g.

(i) Let a and b be two Lie K-algebras with b commutative and $b \xrightarrow{\lambda} g \xrightarrow{\mu} a$ an extension of a by b. For all $x \in g$, the restriction of $\mathrm{ad}_g x$ to b depends only on the class of x modulo b, whence there is an a-module structure on b. Let v be a K-linear mapping of a into g such that $\mu \circ v$ is the identity mapping of a. For $x, y$ in a, we write $f(x, y) = [vx, vy] - v([x, y])$. Show that $f \in Z^2(a, b)$ and that the class c of f in $H^2(a, b)$ does not depend on the choice of v. For two extensions of a by b, which define the same a-module structure on b, to be equivalent, it is necessary and sufficient that the corresponding class $c \in H^2(a, b)$ be the same. For the extension to be inessential, it is necessary and sufficient that $c = 0$. If B is an a-module and $c \in H^2(a, B)$, there exists an extension of a by B (considered as a commutative Lie algebra) defining the given a-module structure on B and the given element c of $H^2(a, B)$.

(j) Let g be a finite-dimensional Lie algebra over K, U its enveloping algebra, h an ideal of g, $\beta$ an invariant bilinear form on g whose restriction to h is non-degenerate, $(y_i)_{1 \leq i \leq n}$ and $(y'_i)_{1 \leq i \leq n}$ two bases of h such that

Finally let $\Gamma$ be the endomorphism of $C^*(g, M)$ extending $t_M$ which maps each cochain f of degree > 0 onto the cochain $t_M \circ f$. Show that $\rho d + d \rho = \Gamma$. (If for $x \in g$ we write:

show first that $a_{kl} = -a'_{lk}$.) Deduce that $\Gamma d = d \Gamma$. If M is simple and finite-dimensional over K, $\beta$ is the associated bilinear form and dim h is not divisible by the characteristic of K, show that $H^p(g, M) = \{0\}$ for all $p$. (Using Proposition 12, show that $\Gamma$ is an automorphism of $C^*(g, M)$ and hence induces automorphisms of $Z^p(g, M)$ and $B^p(g, M)$. For

whence $Z^p(g, M) = B^p(g, M)$.)