D-convolution categories and Hopf algebras

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.

💡 Research Summary

The paper studies three fundamental D‑module categories attached to a smooth affine algebraic group G: the derived category of D‑modules on G itself (D(G)), the derived category on the adjoint quotient stack G/Gₐₙd, and the category of Harish‑Chandra bimodules HC(G). While these categories carry natural convolution monoidal structures, their explicit algebraic descriptions are notoriously difficult, especially the braided monoidal structure on D(G/Gₐₙd) which has only been constructed abstractly in the literature (BD14, Ber17).

The author’s main strategy is to pass to the equivariant (or asymptotic) versions of these categories, denoted D^ℏ(G)_C×, D^ℏ(G/Gₐₙd)_C×, and D(HC^ℏ(G))_C×, where the Rees construction turns filtered algebras into graded ones and a C×‑action records the weight grading. In this setting the work of Beilinson‑Drinfeld on D‑modules and Hecke patterns provides a Koszul duality framework. The author extends this framework to a partial Koszul duality for Harish‑Chandra modules, obtaining three cohomologically graded Hopf algebras:

• A_G, a graded Hopf algebra controlling D^ℏ(G)_C×,
• H_G, a graded Hopf algebra controlling the Harish‑Chandra side,
• A_{G/Gₐₙd}, a graded Hopf algebra controlling D^ℏ(G/Gₐₙd)_C×.

Theorem 1.1 establishes triangulated monoidal equivalences

D^ℏ(G)_C× ≃ D_qs(A_G‑Mod_C×),
D(HC^ℏ(G))_C× ≃ D_qs(H_G‑Mod_C×),
D^ℏ(G/Gₐₙd)_C× ≃ D_qs(A_{G/Gₐₙd}‑Mod_C×),

where D_qs denotes a specific localization of the DG category of modules (the “Koszul‑localized” derived category). These equivalences are proved by combining the classical Koszul dualities of Beilinson‑Drinfeld with the new partial duality for Harish‑Chandra bimodules.

Section 3 constructs the Hopf algebras explicitly. A_G is obtained from the algebra of differential operators on G, identified with the smash product U(g)⋉O(G), equipped with a completed tensor product topology coming from the profinite topology on the linear dual of O(G). A_{G/Gₐₙd} is the strong‑equivariant version, incorporating the adjoint action of G on its Lie algebra. H_G arises from a 1‑shifted Lie bialgebra structure on the Lie algebra g: the author quantizes the 1‑shifted metric Lie algebra D(g) (the double of a 1‑shifted Lie bialgebra) to obtain a Hopf algebra U_δ(D(g))/δ², where δ is a degree‑1 primitive element encoding the cobracket. This construction follows the recent work on 1‑shifted Lie bialgebras (NP25) and provides a physical interpretation in terms of boundary conditions for 2‑dimensional TQFTs.

Section 4 computes the Drinfeld doubles of A_G and H_G. The key result (Theorem 1.3) is that the doubles are twist‑equivalent: the quantum double D(A_G) contains a central primitive element δ* of degree −1, and the quotient D(A_G)/(δ*) is precisely A_{G/Gₐₙd}. Consequently D(A_G) carries an R‑matrix which descends to an R‑matrix on A_{G/Gₐₙd}, endowing the latter with a quasi‑triangular Hopf algebra structure. The existence of a ribbon element θ (Remark 4.8, Proposition A.11) further upgrades the finite‑dimensional module category of A_{G/Gₐₙd} to a pivotal braided monoidal category. When restricted to the heart of the standard t‑structure, this braiding recovers the one constructed by Bezrukavnikov‑Finkelberg‑Ostrik (BFO12). The author argues that the full braiding should match the ∞‑categorical construction of BD14 and Ber17, providing a concrete algebraic model for it.

The paper also offers a physical perspective. The three categories arise as categories of line operators in a 4‑dimensional supersymmetric Yang‑Mills theory; dimensional reduction on a circle yields a 3‑dimensional TQFT whose boundary conditions (Dirichlet, Neumann, and the “cigar” boundary) correspond respectively to the Hopf algebras H_G, A_G, and A_{G/Gₐₙd}. The Drinfeld double of the symmetry algebra of the Dirichlet/Neumann pair reproduces the braided tensor category of line operators on the cigar boundary, confirming predictions from gauge‑theoretic dualities (e.g., Kapustin‑Witten, mirror symmetry). Moreover, the 1‑shifted symplectic structure on the shifted tangent bundle T