nLab
Lie algebra homology

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Dual to Lie algebra cohomology. See there for more.

Definition

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

M U𝔤V(𝔤)=M U𝔤U𝔤 kΛ *𝔤=M kΛ *𝔤. 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, AA in characteristic zero, the Lie algebra homology H (𝔤𝔩(A))H_\bullet(\mathfrak{gl}(A)) of the infinite general linear Lie algebra 𝔤𝔩(A)\mathfrak{gl}(A) with coefficients in AA is, up to a degree shift, the exterior algebra (HC 1(A))\wedge(HC_{\bullet - 1}(A)) on the cyclic homology HC 1(A)HC_{\bullet - 1}(A) of $A:

H (𝔤𝔩(A))(HC 1(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 Pawe l Witkowsk (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

and

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

Last revised on March 26, 2018 at 02:56:33. See the history of this page for a list of all contributions to it.