Hopf algebras are determined by their monoidal derived categories

TL;DR

This paper establishes that finite-dimensional Hopf algebras are uniquely characterized by their monoidal derived categories, linking algebraic gauge equivalence to categorical equivalence.

Contribution

It proves that gauge equivalence of Hopf algebras corresponds precisely to monoidal triangulated equivalence of their derived categories, extending to locally finite tensor categories.

Findings

Gauge equivalent Hopf algebras have monoidal triangulated equivalent derived categories

Monoidal derived equivalences induce monoidal abelian equivalences in tensor categories

Categorical equivalences precisely characterize algebraic gauge relations

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.