Deformation cohomology of Schur-Weyl categories. Free symmetric categories

Davydov, Alexei; Elbehiry, Mohamed

doi:10.48550/arxiv.1908.09192

Cited by 2 publications

(4 citation statements)

References 4 publications

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“…Indeed, in this case we have only two summands in the formula (12) which correspond to two middle paths.…”

Section: Chain Operations Of Low Complexitymentioning

confidence: 99%

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Cosimplicial monoids and deformation theory of tensor categories

Batanin¹,

Davydov²

2020

Preprint

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We introduce a notion of n-commutativity (0 ≤ n ≤ ∞) for cosimplicial monoids in a symmetric monoidal category V, where n = 0 corresponds to just cosimplicial monoids in V, while n = ∞ corresponds to commutative cosimplicial monoids. If V has a monoidal model structure we show (under some mild technical conditions) that the total object of an n-cosimplicial monoid has a natural En+1-algebra structure.Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a 1-commutative cosimplicial monoid and, hence, has an E2-algebra structure similar to the E2-structure on Hochschild complex of an associative algebra provided by Deligne's conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a 2-commutative cosimplicial monoid and, therefore, is naturally an E3-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.

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“…Indeed, in this case we have only two summands in the formula (12) which correspond to two middle paths.…”

Section: Chain Operations Of Low Complexitymentioning

confidence: 99%

“…We also have corresponding odd paths whose action on a ⊗ b we denote by b • i a. Then the formula (12) gives:…”

Section: Chain Operations Of Low Complexitymentioning

confidence: 99%

Cosimplicial monoids and deformation theory of tensor categories

Batanin¹,

Davydov²

2020

Preprint

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“…This theory is the first step to the classification problem of monoidal structures [Dav97] but is also related to quantum algebra and low-dimensional topology. Within this deformation theory, one recovers the category of modules over the quantum group U q (g) as a deformation of the category of modules over the envelopping algebra U (g) of a semisimple Lie algebra g [DE19]. Also, this theory allows to deform the braiding of a tensor category and this can be used to produce link invariants; see [Yet98] where a relation with Vassiliev invariants was established.…”

Section: Introductionmentioning

confidence: 99%

“…A more succesful strategy is to use methods from homological algebra. For instance in [DE19] the DY cohomology of U (g)-mod for a semisimple Lie algebra g has been computed thanks to a natural filtration on this DY complex and using the associated spectral sequence. But in general the DY complex of a tensor category does not admit a natural filtration.…”

Section: Introductionmentioning

confidence: 99%

Davydov--Yetter cohomology and relative homological algebra

Faitg¹,

Gainutdinov²,

Schweigert³

2022

Preprint

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Davydov-Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. We consider such deformations of finite tensor categories and exact tensor functors between them. In [GHS19], DY cohomology with coefficients was introduced and related to the comonad cohomology for a certain adjunction; in the case of tensor categories it reduces to the adjunction for the forgetful functor of the Drinfeld center. We first prove that the DY cohomology groups are isomorphic to the relative Ext groups for this adjunction. From this, we derive the following main results: the vanishing of the first DY cohomology group, long exact sequences of DY cohomology groups which allow to express the groups in terms of Hom spaces, and the existence of a Yoneda-type product on DY cocycles. We apply these results to the category of finite-dimensional modules over a finite-dimensional Hopf algebra and provide a method to compute explicit DY cocycles. We study in detail the examples of the bosonization of exterior algebras ΛC k C[Z 2 ], the Taft algebras and the restricted quantum group of sl 2 at a fourth root of unity Ūi (sl 2 ).

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Deformation cohomology of Schur-Weyl categories. Free symmetric categories

Cited by 2 publications

References 4 publications

Cosimplicial monoids and deformation theory of tensor categories

Cosimplicial monoids and deformation theory of tensor categories

Davydov--Yetter cohomology and relative homological algebra

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