nLab general linear Lie algebra

Definition

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

This is also the endomorphism L-∞ algebra of $V$

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

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

the

• orthogonal Lie algebra

• and special orthogonal Lie algebra.

If $V$ is a complex vector space with an inner product there is

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

the

• unitary Lie algebra

• and special unitary Lie algebra.

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).

References

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.

Lecture notes include

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