nLab
general linear Lie algebra

Context

\infty-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

Contents

Definition

For VV a vector space, the general linear Lie algebra or endomorphism Lie algebra 𝔤𝔩(V)\mathfrak{gl}(V) of VV is the Lie algebra whose elements are linear endomorphisms VVV \to V and whose Lie bracket is given by the commutator of endomorphisms.

This is also the endomorphism L-∞ algebra of VV

If VV is a real vector space that carries an inner product there are the sub-Lie algebras

𝔰𝔬(V)𝔬(V)𝔤𝔩(V) \mathfrak{so}(V) \hookrightarrow \mathfrak{o}(V) \hookrightarrow \mathfrak{gl}(V)

the

If VV is a complex vector space with an inner product there is

𝔰𝔬(V)𝔬(V)𝔤𝔩(V) \mathfrak{so}(V) \hookrightarrow \mathfrak{o}(V) \hookrightarrow \mathfrak{gl}(V)

the

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).

References

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.

Lecture notes include

  • Jean-Louis Loday, Cyclic Homology Theory, Part II, notes taken by Pawe l Witkowsk (2007) (pdf)

Last revised on March 26, 2018 at 02:54:07. See the history of this page for a list of all contributions to it.