Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

Kobyzev, Ivan; Shapiro, Ilya

doi:10.3842/sigma.2018.098

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…More precisely, in Section 9.1 we attempt to explain, by sketching the commutative case, why the mixed anti-Yetter-Drinfeld contramodules in the Hopf setting are exactly analogous to the D-modules in the geometric setting. In Section 9.2 we link the monadic approach with the previous attempts [10,15,14,18,19] to understand Hopf cyclic coefficients as centers of certain bimodule categories. that are compatible with the S 1 -actions.…”

Section: Introductionmentioning

confidence: 99%

Categorified Chern character and cyclic cohomology

Shapiro

2019

Preprint

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We examine Hopf cyclic cohomology in the same context as the analysis [1,2,3] of the geometry of loop spaces LX in derived algebraic geometry and the resulting close relationship between S 1 -equivariant quasi-coherent sheaves on LX and D X -modules. Furthermore, the Hopf setting serves as a toy case for the categorification of Chern character theory as discussed in [20]. More precisely, this examination naturally leads to a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that of the usual mixed complexes for the trivial Hopf algebra and generalizes the notion of stable anti-Yetter-Drinfeld contramodules that have thus far served as the coefficients for Hopf-cyclic theories [4]. The cohomology is then obtained as a Hom in this dgcategory between a Chern character object associated to an algebra and an arbitrary coefficient mixed anti-Yetter-Drinfeld contramodule.

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Section: Introductionmentioning

confidence: 99%

Categorified Chern character and cyclic cohomology

Shapiro

2019

Preprint

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“…The Sections 6 and 7 address the definitions of Hopf algebroids, the biclosed property of their categories of modules (we provide a direct proof, but the fact can also be obtained from [21]) and finally the translation into formulas of the definition of anti-Yetter-Drinfeld contramodules and their stability. We remark that we handled the case of anti-Yetter-Drinfeld module analogues in our follow-up paper [14]. We did not want to increase the length of the exposition any further and modules, despite being more familiar, are actually technically more difficult and less natural/suitable from the categorical perspective on the subject of coefficients.…”

Section: Introductionmentioning

confidence: 99%

A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

Kobyzev

Shapiro

2018

Appl Categor Struct

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We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules for Hopf algebras.

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Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

Cited by 2 publications

References 17 publications

Categorified Chern character and cyclic cohomology

Categorified Chern character and cyclic cohomology

A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

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