D-convolution categories and Hopf algebras
Wenjun Niu
TL;DR
The paper develops a Hopf-algebraic framework for monoidal D-module categories associated to a smooth affine group $G$, translating convolution-based structures on ${\mathcal{D}}(G)$, ${\mathcal{D}}(G/G_{ad})$, and $D({\mathrm{HC}}(G))$ into localized modules over graded Hopf algebras $\mathcal{A}_G$, $A_{G/G_{ad}}$, and $H_G$. By passing to equivariant, cohomologically graded settings ${\mathcal{D}}_\hbar(\cdot)^{\mathbb{C}^\times}$ and applying Beilinson-Drinfeld Koszul dualities, the authors establish monoidal equivalences with DG-module localizations $D_{qs}(\cdot-\mathrm{Mod}^{\mathbb{C}^\times})$, enabling an explicit braided monoidal structure on ${\mathcal{D}}_\hbar(G/G_{ad})^{\mathbb{C}^\times}$ whose heart recovers the Bezrukavnikov-Finkelberg-Ostrik braiding. The construction is physically motivated by 3d TQFTs and 1-shifted Lie bialgebras, with the Hopf algebras $H_G$, $\mathcal{A}_G$, and $A_{G/G_{ad}}$ encoding symmetry data and their doubles furnishing the braided center, all complemented by an analysis of doubles and an appendix on Drinfeld doubles for infinite-dimensional cases. This provides a concrete, computable bridge between geometric representation theory and quantum algebra, enabling explicit braiding data and center-like structures for D-modules on group stacks. The work thus advances a Hopf-algebraic paradigm for D-convolution categories with potential applications to geometric Langlands and related quantum-field-theoretic interpretations.
Abstract
For a smooth affine algebraic group $G$, one can attach various D-module categories to it that admit convolution monoidal structure. We consider the derived category of D-modules on $G$, the stack $G/G_{ad}$ and the category of Harish-Chandra bimodules. Combining the work of Beilinson-Drinfeld on D-modules and Hecke patterns with the recent work of the author with Dimofte and Py, we show that each of the above categories (more precisely the equivariant version) is monoidal equivalent to a localization of the DG category of modules of a graded Hopf algebra. As a consequence, we give an explicit braided monoidal structure to the derived category of D-modules on $G/G_{ad}$, which when restricted to the heart, recovers the braiding of Bezrukavnikov-Finkelberg-Ostrik.