Finite dimensional Lie algebras with trivial homology

Question

Are there any known examples of Lie algebras $\mathfrak{g}$ such $H_*(\mathfrak{g})=0$ for all $*>0?$ Better still are there such algebras that this condition holds for arbitrary coefficient modules and not just for the trivial one?

I think the universal enveloping algebra of $\mathfrak{g}$ should be semisimple. Outside of this, I don't know anything. I don't know even how to start answering the question.

Edit: Correcting the initial ambiguity of my question, I assume that $\mathfrak{g}$ is finite dimensional.

Answer 1

If $\mathfrak{g}$ has dimension $n<\infty$, then

• either $\mathfrak{g}$ is unimodular (i.e., the linear form $T:x\mapsto\mathrm{Tr}(\mathrm{ad}(x))$ vanishes), and then $H_n(\mathfrak{g})\neq\{0\}$ (it is 1-dimensional)
• or $\mathfrak{g}$ is not unimodular, and then $T$ is a nonzero homomorphism from $\mathfrak{g}$ to the ground field, so $H_1(\mathfrak{g})\neq 0$.

In characteristic zero we have another alternative: if $\mathfrak{g}\neq\{0\}$:

• either $\mathfrak{g}$ is solvable, and then $H_1(\mathfrak{g})\neq 0$;
• or $\mathfrak{g}$ is not solvable, and then $H_3(\mathfrak{g})\neq 0$. Indeed, $\mathfrak{g}$ admits a nonzero semisimple Lie algebra as a retract (Levi factor), and the result follows ($\ast$) from the nonvanishing of $H_3(\mathfrak{g})$ when $\mathfrak{g}$ is simple (Chevalley-Eilenberg).

($\ast$) Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{h}$ a retract of $\mathfrak{g}$ (i.e., a subalgebra such that there exists a homomorphism $p:\mathfrak{g}\to\mathfrak{h}$ such that $p\circ i=\mathrm{id}_\mathfrak{h}$, where $i$ is the inclusion. Then the canonical map $i_*:H_n(\mathfrak{h})\to H_n(\mathfrak{g})$ is injective for all $n$. Proof: Since $H_n(-)$ is functorial, we have $p_*\ast i_*$ equal to the identity of $H_n(\mathfrak{h})$, so that $i_*$ is injective.

Answer 2

For finite-dimensional semisimple Lie algebras we have the following result by Whitehead.

Theorem (Whitehead): If L is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and $M$ is a nontrivial irreducible module, then $$ H^i(L,M)=H_i(L,M)=0 $$ for all $i\ge 0$.

For the trivial module Whitehead's result is not true. We have $H^1(L)=H^2(L)=0$, but $H^3(L)\neq 0$ by a result of Chevalley and Eilenberg in $1948$. By Poincare-duality (semisimple Lie algebras are unimodular) we also obtain a non-zero third homology group.