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Semisimple weak Hopf algebras

Dmitri Nikshych

TL;DR

The paper develops the theory of semisimple weak Hopf algebras, extending classical results from ordinary semisimple Hopf algebras to the weak setting. It establishes criteria for semisimplicity via integrals, analyzes the square of the antipode $S^2$, and relates Frobenius-Perron dimensions to an inclusion matrix $\Lambda$, including trace forms and the class equation. It also extends Larson–Radford trace formulas and proves divisibility-type results for dimensions of module algebras, illuminating the structure of $A$-modules and (co)module algebras within fusion-category frameworks. These results pave the way toward a classification program for semisimple weak Hopf algebras and their fusion-category realizations, with implications for operator algebras and quantum group theory.

Abstract

We develop the theory of semisimple weak Hopf algebras and obtain analogues of a number of classical results for ordinary semisimple Hopf algebras. We prove a criterion for semisimplicity and analyze the square of the antipode S^2 of a semisimple weak Hopf algebra A. We explain how the Frobenius-Perron dimensions of irreducible A-modules and eigenvalues of S^2 can be computed using the inclusion matrix associated to A. A trace formula of Larson and Radford is extended to a relation between the global and Frobenius-Perron dimensions of A. Finally, an analogue of the Class Equation of Kac and Zhu is established and properties of $A$-module algebras and their dimensions are studied.

Semisimple weak Hopf algebras

TL;DR

The paper develops the theory of semisimple weak Hopf algebras, extending classical results from ordinary semisimple Hopf algebras to the weak setting. It establishes criteria for semisimplicity via integrals, analyzes the square of the antipode , and relates Frobenius-Perron dimensions to an inclusion matrix , including trace forms and the class equation. It also extends Larson–Radford trace formulas and proves divisibility-type results for dimensions of module algebras, illuminating the structure of -modules and (co)module algebras within fusion-category frameworks. These results pave the way toward a classification program for semisimple weak Hopf algebras and their fusion-category realizations, with implications for operator algebras and quantum group theory.

Abstract

We develop the theory of semisimple weak Hopf algebras and obtain analogues of a number of classical results for ordinary semisimple Hopf algebras. We prove a criterion for semisimplicity and analyze the square of the antipode S^2 of a semisimple weak Hopf algebra A. We explain how the Frobenius-Perron dimensions of irreducible A-modules and eigenvalues of S^2 can be computed using the inclusion matrix associated to A. A trace formula of Larson and Radford is extended to a relation between the global and Frobenius-Perron dimensions of A. Finally, an analogue of the Class Equation of Kac and Zhu is established and properties of -module algebras and their dimensions are studied.

Paper Structure

This paper contains 21 sections, 30 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition and basic properties of weak Hopf algebras
  4. Basic properties of weak Hopf algebras
  5. Rigid monoidal categories and fusion categories
  6. Representation category of a weak Hopf algebra and reconstruction of fusion categories
  7. Grothendieck rings and modules and Frobenius-Perron dimensions
  8. Integrals, semisimplicity, and dimension theory
  9. Integrals in weak Hopf algebras
  10. Group-like and pivotal elements in a weak Hopf algebra
  11. Dimension theory for weak Hopf algebras
  12. Implementation of the antipode and pseudo-unitary weak Hopf algebras
  13. The class equation
  14. The character algebra and the Grothendieck ring of a weak Hopf algebra
  15. The class equation for weak Hopf algebras
  16. ...and 6 more sections

Key Result

Lemma 2.5.3

The above map $\psi$ is a unique, up to a positive scalar multiple, $K_0(A)$-module homomorphism from $K_0(M)$ to $\mathbb{R}$ with the property $\psi(\xi_i)>0$ for all $i=1,\dots t$.

Theorems & Definitions (84)

  • Definition 2.1.1: BSz1BNSz
  • Remark 2.1.2
  • Remark 2.1.3
  • Remark 2.1.4
  • Definition 2.2.2
  • Remark 2.2.3
  • Definition 2.3.1
  • Definition 2.4.1
  • Remark 2.4.2
  • Definition 2.5.1
  • ...and 74 more