nLab Lie algebra homology

Idea

Dual to Lie algebra cohomology. See there for more.

Definition

The abelian homology of a $k$-Lie algebra $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H_*^{Lie}(\mathfrak{g},M) = Tor^{U\mathfrak{g}}_*(k,M)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U\mathfrak{g}$. In particular it is a derived functor. It can be computed using Chevalley-Eilenberg chain complex $V(\mathfrak{g})$ as the homology of the chain complex

$$M \otimes_{U\mathfrak{g}} V(\mathfrak{g}) = M\otimes_{U\mathfrak{g}} U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g} = M\otimes_k \Lambda^* \mathfrak{g}.$$

Properties

Loday-Quillen-Tsygan theorem

The Loday-Quillen-Tsygan theorem (Loday-Quillen 84, Tsygan 83) states that for any associative algebra, $A$ in characteristic zero, the Lie algebra homology $H_\bullet(\mathfrak{gl}(A))$ of the infinite general linear Lie algebra $\mathfrak{gl}(A)$ with coefficients in $A$ is, up to a degree shift, the exterior algebra $\wedge(HC_{\bullet - 1}(A))$ on the cyclic homology $HC_{\bullet - 1}(A)$ of $A$:

$$H_\bullet(\mathfrak{gl}(A)) \;\simeq\; \wedge( HC_{\bullet - 1}(A) )$$

(see e.g Loday 07, theorem 1.1).

Lecture notes include

• Jean-Louis Loday, Cyclic Homology Theory, Part II, notes taken by Paweł Witkowski (2007) (pdf)

References

• C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63, (1948). 85–124.

The Loday-Quillen-Tsygan theorem is originally due, independently, to

• Jean-Louis Loday, Daniel Quillen, Cyclic homology and the Lie algebra homology of matrices Comment. Math. Helv., 59(4):569–591, 1984.

and

• Boris Tsygan, Homology of matrix algebras over rings and the Hochschild homology, Uspeki Math. Nauk., 38:217–218, 1983.