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Constructing Modular Tensor Categories with Hopf Algebras

Kun Zhou

Abstract

A modular tensor category is a non-degenerate ribbon finite tensor category. And a ribbon factorizable Hopf algebra is exactly the Hopf algebra whose finite-dimensional representations form a modular tensor category. The goal of this paper is to construct both semisimple and non-semisimple modular categories with Hopf algebras. In particular, we study central extensions of Hopf algebras and characterize some conditions for a quotient Hopf algebra to be a factorizable one. Then we give a special way to obtain ribbon elements. As a result, we apply to some finite dimensional pointed Hopf algebras introduced by Andruskiewitsch etc and obtain two families of ribbon factorizable Hopf algebras. With little restrictions to their parameters, we prove that their representation categories are prime modular tensor categories which are not tensor equivalent to representation categories of small quantum groups. Lastly, we construct a family of semisimple ribbon factorizable Hopf algebas A(p,q), where p,q are prime numbers satisfying some conditions. And we decompose Rep(A(p,q)) into prime modular tensor categories and prove that A(p,q) can't be obtained by some obvious ways, i.e. it's not tensor product of trivial Hopf algebras (group algebras or their dual) and Drinfeld doubles.

Constructing Modular Tensor Categories with Hopf Algebras

Abstract

A modular tensor category is a non-degenerate ribbon finite tensor category. And a ribbon factorizable Hopf algebra is exactly the Hopf algebra whose finite-dimensional representations form a modular tensor category. The goal of this paper is to construct both semisimple and non-semisimple modular categories with Hopf algebras. In particular, we study central extensions of Hopf algebras and characterize some conditions for a quotient Hopf algebra to be a factorizable one. Then we give a special way to obtain ribbon elements. As a result, we apply to some finite dimensional pointed Hopf algebras introduced by Andruskiewitsch etc and obtain two families of ribbon factorizable Hopf algebras. With little restrictions to their parameters, we prove that their representation categories are prime modular tensor categories which are not tensor equivalent to representation categories of small quantum groups. Lastly, we construct a family of semisimple ribbon factorizable Hopf algebas A(p,q), where p,q are prime numbers satisfying some conditions. And we decompose Rep(A(p,q)) into prime modular tensor categories and prove that A(p,q) can't be obtained by some obvious ways, i.e. it's not tensor product of trivial Hopf algebras (group algebras or their dual) and Drinfeld doubles.
Paper Structure (12 sections, 34 theorems, 81 equations)

This paper contains 12 sections, 34 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modular tensor categories and ribbon factorizable Hopf algebras.
  4. Hopf exact sequence.
  5. The pointed Hopf algebras $u(\mathcal{D},\lambda,\mu)$.
  6. Nichols algebras and the Hopf algebras $\mathcal{U}(\mathcal{D}_{red})$.
  7. Main way to construct ribbon factorizable Hopf algebras
  8. Some applications
  9. Ribbon factorizable Hopf algebras $D(u(\mathcal{D},\lambda,\mu))/I$.
  10. Ribbon factorizable Hopf algebras $D(\mathcal{U}(\mathcal{D}_{red}))/I$.
  11. Property of prime and comparison.
  12. Prime modular tensor categories $\operatorname{Rep}(D(G_{m,n}))$ and semisimple ribbon factorizable Hopf algebras $A(p,q)$.

Key Result

Theorem 2.3

Let $(\mathcal{D}, \otimes, \mathrm{1}, c)$ be a non-degenerate finite braided tensor category, let $\mathcal{E}$ be a topologizing braided tensor subcategory of $\mathcal{D}$. Then $C_\mathcal{D}(\mathcal{E})$ is non-degenerate braided tensor category if and only if $\mathcal{E}$ is non-degenerate.

Theorems & Definitions (79)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 69 more