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Hopf algebroids and Grothendieck-Verdier duality

Robert Allen

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.

Hopf algebroids and Grothendieck-Verdier duality

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.
Paper Structure (9 sections, 7 theorems, 41 equations)

This paper contains 9 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Hopf algebroids
  3. Grothendieck-Verdier categories
  4. Known results
  5. Outline
  6. Hopf algebroids
  7. Closed structure
  8. Duality structure
  9. Acknowledgements

Key Result

Proposition 3.2

Let $M$, $N$ be $H$-modules. The category of $H$-modules is right and left closed, with with left action of $H$ given by right multiplication on $H$ in $H \otimes_A M$ and $M \otimes_A H$, where the tensor products are in the category of $H$-modules.

Theorems & Definitions (22)

  • Definition 2.1: BS04
  • Definition 2.2: BS04
  • Remark
  • Remark
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 12 more