nLab Loday-Quillen-Tsygan theorem

Contents

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

• 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.

See also additive K-theory and

• 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)

See also

• Jean-Louis Loday, Claudio Procesi, Cyclic homology and lambda-operations, In: Frederick Jardine , Victor Snaith (eds.) Algebraic K-Theory: Connections with Geometry and Topology NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 279. Springer, Dordrecht (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:

• Grégory Ginot, Owen Gwilliam, Alastair Hamilton, Mahmoud Zeinalian, Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem, Adv. Math. 409A (2022) 108631 [arXiv:2108.12109, doi:10.1016/j.aim.2022.108631]