# nLab Loday-Quillen-Tsygan theorem

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

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)

Some extensions:

• Atabey Kaygun, Loday–Quillen–Tsygan Theorem for coalgebras (arXiv:math/0411661)

• Lukas Miaskiwskyi, Continuous cohomology of gauge algebras and bornological Loday-Quillen-Tsygan theorems, arXiv:2206.08879

• Masoud Khalkhali, Homology of $L_{\infty}$-algebras and cyclic homology, arXiv:9805052

• Benjamin Hennion, The tangent complex of K-theory, Journal de l’École polytechnique — Mathématiques 8 (2021) 895–932.

On a kind of BV-quantization of the Loday-Quillen-Tsygan theorem and relating to the large $N$-limit of Chern-Simons theory:

Last revised on March 4, 2023 at 06:07:04. See the history of this page for a list of all contributions to it.