Hopf algebroids and Grothendieck-Verdier duality

TL;DR

This paper explores how Grothendieck-Verdier duality, a key structure in monoidal categories, extends to categories of modules over Hopf algebroids, generalizing concepts of rigidity and duality.

Contribution

It demonstrates that finite-dimensional modules over Hopf algebroids with bijective antipodes inherit Grothendieck-Verdier duality from bimodule categories, extending duality theory.

Findings

Categories of modules over Hopf algebroids inherit Grothendieck-Verdier duality.

The duality structure is connected to the algebraic properties of the base algebra.

The work generalizes rigidity concepts to non-commutative settings.

Abstract

Grothendieck-Verdier duality is a powerful and ubiquitous structure on monoidal categories, which generalises the notion of rigidity. Hopf algebroids are a generalisation of Hopf algebras, to a non-commutative base ring. Just as the category of finite-dimensional modules over a Hopf algebra inherits rigidity from the category of vector spaces, we show that the category of finite-dimensional modules over a Hopf algebroid with bijective antipode inherits a Grothendieck-Verdier structure from the category of bimodules over its base algebra. We investigate the algebraic and categorical structure of this duality.