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Hopf crossed module (co)algebras

Kursat Sozer, Alexis Virelizier

Abstract

Given a crossed module $χ$, we introduce Hopf $χ$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of representations are monoidal and $χ$-graded (meaning that both objects and morphisms have degrees which are related via $χ$).

Hopf crossed module (co)algebras

Abstract

Given a crossed module , we introduce Hopf -(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of representations are monoidal and -graded (meaning that both objects and morphisms have degrees which are related via ).
Paper Structure (69 sections, 12 theorems, 137 equations)

This paper contains 69 sections, 12 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Categorical preliminaries
  3. Conventions on monoidal categories
  4. Braided and symmetric categories
  5. Cartesian monoidal categories
  6. Rigid categories
  7. Closed monoidal categories
  8. Pivotal categories
  9. Penrose graphical calculus
  10. Categorical Hopf algebras
  11. Categorical algebras
  12. Categorical coalgebras
  13. Categorical bialgebras
  14. Categorical Hopf algebras
  15. Examples
  16. ...and 54 more sections

Key Result

Lemma 7.1

Let $A=\{A_x\}_{x \in H}$ be a Hopf $\chi$-coalgebra, with antipode $S=\{S_x\}_{x \in H}$ and $\chi$-action $\phi=\{\phi_{x,e} \}_{(x,e) \in H \times E}$. Then for all $x \in H$ and $e \in E$.

Theorems & Definitions (22)

  • Lemma 7.1
  • proof
  • Lemma 7.2
  • proof
  • Theorem 8.1
  • proof
  • Corollary 8.2
  • proof
  • Theorem 8.3
  • proof
  • ...and 12 more