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Pivotality, twisted centres and the anti-double of a Hopf monad

Sebastian Halbig, Tony Zorman

Abstract

Finite-dimensional Hopf algebras admit a correspondence between so-called pairs in involution, one-dimensional anti-Yetter--Drinfeld modules and algebra isomorphisms between the Drinfeld and anti-Drinfeld double. We extend it to general rigid monoidal categories and provide a monadic interpretation under the assumption that certain coends exist. Hereto we construct and study the anti-Drinfeld double of a Hopf monad. As an application the connection with the pivotality of Drinfeld centres and their underlying categories is discussed.

Pivotality, twisted centres and the anti-double of a Hopf monad

Abstract

Finite-dimensional Hopf algebras admit a correspondence between so-called pairs in involution, one-dimensional anti-Yetter--Drinfeld modules and algebra isomorphisms between the Drinfeld and anti-Drinfeld double. We extend it to general rigid monoidal categories and provide a monadic interpretation under the assumption that certain coends exist. Hereto we construct and study the anti-Drinfeld double of a Hopf monad. As an application the connection with the pivotality of Drinfeld centres and their underlying categories is discussed.
Paper Structure (27 sections, 55 theorems, 180 equations, 2 figures)

This paper contains 27 sections, 55 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Monoidal categories, bimodule categories and the centre construction
  3. From monoidal to pivotal categories
  4. Monoidal categories, functors, and natural transformations
  5. Rigidity and pivotality
  6. Braided categories
  7. Bimodule categories and the centre construction
  8. Left right and bimodule categories
  9. The Drinfeld centre of a monoidal category
  10. Heaps
  11. Pivotal structures and twisted centres
  12. Twisted centres and their Picard heaps
  13. Quasi-pivotality
  14. Pivotality of the Drinfeld centre
  15. Bimonads and comodule monads as coordinate systems for (twisted) centres
  16. ...and 12 more sections

Key Result

theorem 1

For any finite-dimensional Hopf algebra $H$ the following statements are equivalent:

Figures (2)

  • Figure 1: Various types of monoidal and module categories, as well as (some) relations between them.
  • Figure 2: A cobweb of adjunctions, monads and various versions of the Drinfeld and anti-Drinfeld centre.

Theorems & Definitions (141)

  • theorem 1: Halbig2019
  • theorem 2
  • theorem 3
  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7: castelnuovo1905
  • ...and 131 more