An adjunction theorem for Davydov-Yetter cohomology and infinitesimal braidings - Centre International de Mathématiques et d'Informatique de Toulouse
Pré-Publication, Document De Travail Année : 2024

An adjunction theorem for Davydov-Yetter cohomology and infinitesimal braidings

Théorème d'adjonction pour la cohomologie de Davydov-Yetter et tressages infinitésimaux

Résumé

Davydov-Yetter cohomology $H_{\mathrm{DY}}^{\bullet}(F)$ is associated to a monoidal functor $F: \mathcal{C} \to \mathcal{D}$ between $\Bbbk$-linear monoidal categories where $\Bbbk$ is a field, and its second degree classifies the infinitesimal deformations of the monoidal structure of $F$. Our main result states that if $F$ admits a right adjoint $R$, then there is an object $Γ$ in the Drinfeld center $\mathcal{Z}(\mathcal{C})$ defined in terms of $R$ such that the Davydov-Yetter cohomology of $F$ can be expressed as the Davydov-Yetter cohomology of the identity functor on $\mathcal{C}$ with the coefficient $Γ$. We apply this result in the case when the product functor $\otimes: \mathcal{C} \boxtimes\mathcal{C} \to\mathcal{C}$ has a monoidal structure given by a braiding $c$ on $\mathcal{C}$ and determine explicitly the coefficient $Γ$ as a coend object in $\mathcal{Z}(\mathcal{C}) \boxtimes \mathcal{Z}(\mathcal{C})$. The motivation is that $H^{\bullet}_{\mathrm{DY}}(\otimes)$ contains a ``space of infinitesimal braidings tangent to $c$'' in a way that we describe precisely. For $\mathcal{C} = H\text{-}\mathrm{mod}$, where $H$ is a finite-dimensional Hopf algebra over a field $\Bbbk$, this is the Zariski tangent space to the affine variety of R-matrices for $H$. In the case of perfect $\Bbbk$, we give a dimension formula for this space as an explicit end involving only (low-degree) relative Ext's of the standard adjunction between $\mathcal{Z}(\mathcal{C})$ and $\mathcal{C}$. As a further application of the adjunction theorem, we describe deformations of the restriction functor associated to a Hopf subalgebra and a Drinfeld twist. Both applications are illustrated in the example of bosonization of exterior algebras.
Fichier principal
Vignette du fichier
2411.19111v1.pdf (1.14 Mo) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04878828 , version 1 (10-01-2025)

Identifiants

Citer

Matthieu Faitg, Azat M. Gainutdinov, Christoph Schweigert. An adjunction theorem for Davydov-Yetter cohomology and infinitesimal braidings. 2024. ⟨hal-04878828⟩
0 Consultations
0 Téléchargements

Altmetric

Partager

More