Hopf-cyclic homology and cohomology with coefficients

By P. M. Hajac, Masoud Khalkhali, B. Rangipour, Yorck Sommerhäuser. 2003

130 citations recorded by the source corpus.

Abstract Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter–Drinfeld compatibility condition leading to the concept of a stable anti-Yetter–Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/964a58b245ac9032a17ca92e15da913815be0ab2

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In Algebra and Number Theory

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Citations in the corpus

Lectures on Noncommutative Geometry (2005, background): “These modules are noncommutative coefficient systems for the general Hopf cyclic cohomology theory developed in [68].”