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.
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