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Loday-Quillen-Tsygan theorem

Context

Algebra

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

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Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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 AA:

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 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.

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, C. 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)

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

Last revised on March 26, 2018 at 11:28:17. See the history of this page for a list of all contributions to it.