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The Hopf category of Frobenius algebras

Paul Großkopf, Joost Vercruysse

Abstract

We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of Batista-Caenepeel-Vercruysse), and the universal comeasuring algebras between Frobenius algebras turn the category Frobenius algebras into a Hopf opcategory. We also discuss duality and compatibility results between these structures. Our theory vastly generalizes the well-known fact that any homomorphism beween Frobenius algebras is an isomorphism, but also allows to go beyond classical (iso)morphisms between Frobenius algebras, especially in finite characteristic, as we show by some explicit examples. The paper is concluded with some open questions and considerations about Topological Quantum Field Theories.

The Hopf category of Frobenius algebras

Abstract

We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of Batista-Caenepeel-Vercruysse), and the universal comeasuring algebras between Frobenius algebras turn the category Frobenius algebras into a Hopf opcategory. We also discuss duality and compatibility results between these structures. Our theory vastly generalizes the well-known fact that any homomorphism beween Frobenius algebras is an isomorphism, but also allows to go beyond classical (iso)morphisms between Frobenius algebras, especially in finite characteristic, as we show by some explicit examples. The paper is concluded with some open questions and considerations about Topological Quantum Field Theories.

Paper Structure

This paper contains 15 sections, 21 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and first results
  3. Morphisms of Frobenius algebras
  4. Hopf categories and Hopf opcategories
  5. Measuring and comeasuring semi-Hopf categories for $\Omega$-algebras
  6. Measuring coalgebras
  7. Comeasuring algebras
  8. The Hopf category of Frobenius algebras
  9. Dualities
  10. Sweedler dual of a semi-Hopf opcategory
  11. Duality for Frobenius algebras and the antipode of their universal Hopf category
  12. Examples
  13. Conclusions and outlook
  14. Remaining questions
  15. Motivation from topological quantum field theory

Key Result

Proposition 2.2

Every morphism of Frobenius algebras is an isomorphism.

Theorems & Definitions (60)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 50 more