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Davydov-Yetter cohomology and relative homological algebra

Matthieu Faitg, Azat M. Gainutdinov, Christoph Schweigert

Abstract

Davydov--Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category $\mathcal{C}$ are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ relative to $\mathcal{C}$. From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of $\mathcal{Z}(\mathcal{C})$. Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras $Λ\mathbb{C}^k \rtimes \mathbb{C}[\mathbb{Z}_2]$, the Taft algebras and the small quantum group of $\mathfrak{sl}_2$ at a root of unity.

Davydov-Yetter cohomology and relative homological algebra

Abstract

Davydov--Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center relative to . From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of . Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras , the Taft algebras and the small quantum group of at a root of unity.
Paper Structure (35 sections, 48 theorems, 280 equations, 2 figures)

This paper contains 35 sections, 48 theorems, 280 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relative homological algebra
  3. Resolvent pairs and relative $\mathop{\mathrm{Ext}}\nolimits$ groups
  4. Properties of relative $\mathop{\mathrm{Ext}}\nolimits$ groups
  5. Yoneda description of relative $\mathop{\mathrm{Ext}}\nolimits$ groups
  6. The Yoneda product
  7. The long exact sequence for relative $\mathop{\mathrm{Ext}}\nolimits$ groups
  8. Relatively projective cover
  9. Comonad cohomology and relative $\mathop{\mathrm{Ext}}\nolimits$ groups
  10. Relative $\mathop{\mathrm{Ext}}\nolimits$ groups for tensor categories
  11. Monoidal resolvent pairs
  12. Tensor product of algebras in braided tensor categories
  13. Davydov--Yetter cohomology and relative $\mathop{\mathrm{Ext}}\nolimits$ groups
  14. The centralizer of a functor and the associated adjunction
  15. Relation to Davydov-Yetter cohomology
  16. ...and 20 more sections

Key Result

Theorem 1

The Davydov--Yetter cohomology of an exact tensor functor $F : \mathcal{C} \to \mathcal{D}$ with coefficients $\mathsf{V}, \mathsf{W} \in \mathcal{Z}(F)$ is isomorphic to the relative Ext groups of the adjunction defined by the forgetful functor $\mathcal{Z}(F) \to \mathcal{D}$: for all $n \geq 0$.

Figures (2)

  • Figure 1: Proof of the commutation of the diagram \ref{['diagramPsiAndHalfBraidings']}.
  • Figure 2: Proof of Lemma \ref{['lemmaDYYonedaProductFor1Cocycle']}

Theorems & Definitions (101)

  • Theorem 1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 91 more