# nLab Loday-Quillen-Tsygan theorem

### Context

#### Algebra

higher algebra

universal algebra

cohomology

## Idea

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 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.

• Boris Tsygan, Boris Feigin, Additive K-theory, in K-theory, arithmetic and geometry, LNM 1289 (1987), edited by Yu. I. Manin, pp. 67–209, seminar 1984-1986 in Moscow), MR89a:18017

Lecture notes include

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