Hopf algebras are determined by their monoidal derived categories
Yuying Xu, Junhua Zheng
TL;DR
The paper establishes a reconstruction framework showing that gauge equivalence of finite-dimensional Hopf algebras $H,H'$ is equivalent to monoidal abelian and monoidal triangulated equivalences of their module and derived categories, respectively, and extends the result to locally finite tensor Grothendieck categories. It develops the theory of monoidal t-structures, introduces tensor reducedness, and proves that hearts of tensor reduced monoidal t-structures are monoidal abelian (and often tensor) categories, enabling transfer of monoidal structure through derived equivalences. A key outcome is a three-way equivalence: equivalences at the level of finitely presented subcategories, ambient categories, and their bounded derived categories, all in the monoidal setting. The authors also provide a Rickard-type stable-equivalence analogue in the monoidal context, applying it to Hopf algebras and weak quasi-Hopf algebras, which yields a robust bridge between abelian, triangulated, and stable monoidal structures with potential impact on classification problems in quantum algebra. Overall, the work offers a unified approach to “reconstruction” results in tensor settings, linking algebraic gauge data to derived-category invariants.
Abstract
We show that two finite-dimensional Hopf algebras are gauge equivalent if and only if their bounded derived categories are monoidal triangulated equivalent. More generally, a monoidal derived equivalence between locally finite tensor Grothendieck categories induces a monoidal abelian equivalence.