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Constructing non-semisimple modular categories with local modules

Robert Laugwitz, Chelsea Walton

TL;DR

The paper extends modular tensor category theory to non-semisimple settings by introducing rigid Frobenius algebras in braided finite tensor categories and proving that their local modules form modular categories. It develops a robust framework linking local modules to relative monoidal centers, enabling construction of new non-semisimple modular categories and giving FP-dimension bounds. It also provides concrete classifications and examples, including rigid Frobenius algebras in Drinfeld centers and Yetter–Drinfeld contexts, and discusses anisotropy and Witt-equivalence questions relevant to topological phases and quantum algebra. The results generalize known semisimple constructions, offering new avenues for non-semisimple modular invariants and boundary phenomena in topological quantum field theories.

Abstract

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171 (2002), no. 2] in the semisimple setup. Examples of non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided. In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification by A. Davydov [J. Algebra 323 (2010), no. 5] to arbitrary characteristic.

Constructing non-semisimple modular categories with local modules

TL;DR

The paper extends modular tensor category theory to non-semisimple settings by introducing rigid Frobenius algebras in braided finite tensor categories and proving that their local modules form modular categories. It develops a robust framework linking local modules to relative monoidal centers, enabling construction of new non-semisimple modular categories and giving FP-dimension bounds. It also provides concrete classifications and examples, including rigid Frobenius algebras in Drinfeld centers and Yetter–Drinfeld contexts, and discusses anisotropy and Witt-equivalence questions relevant to topological phases and quantum algebra. The results generalize known semisimple constructions, offering new avenues for non-semisimple modular invariants and boundary phenomena in topological quantum field theories.

Abstract

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171 (2002), no. 2] in the semisimple setup. Examples of non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided. In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification by A. Davydov [J. Algebra 323 (2010), no. 5] to arbitrary characteristic.
Paper Structure (26 sections, 44 theorems, 135 equations)

This paper contains 26 sections, 44 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Monoidal Categories
  3. Monoidal categories and monoidal functors
  4. Rigid and pivotal monoidal categories, and quantum dimension
  5. Finite tensor categories and Frobenius--Perron dimension
  6. Braided tensor categories, the monoidal center Z(C), and the Müger center C'
  7. Ribbon tensor categories
  8. Non-degeneracy and modular tensor categories
  9. Preliminaries on algebraic structures in braided finite tensor categories
  10. Algebras, coalgebras, and their (co)modules
  11. Bialgebras and Hopf algebras, and their (co)modules
  12. Frobenius algebras
  13. Modularity of categories of local modules
  14. Categories of modules over (braided) commutative algebras
  15. Local modules
  16. ...and 11 more sections

Key Result

Proposition 1.2

Let $\cal{C}$ be a ribbon finite tensor category (e.g., a modular tensor category), and take $A$ an algebra in $\cal{C}$. Then the following statements are equivalent.

Theorems & Definitions (102)

  • Definition 1.1: Definition \ref{['def:rigid-Frob']}
  • Proposition 1.2: Proposition \ref{['prop:conn-etale']}
  • Theorem 1.3: Theorem \ref{['thm:locmodular']}
  • Theorem 1.4: Theorem \ref{['thm:ZRepA']}
  • Corollary 1.5: Corollary \ref{['cor:local-mod']}
  • Proposition 1.6: Proposition \ref{['prop:local-triv']}
  • Proposition 1.7: Proposition \ref{['prop:ex-charp']}
  • Theorem 1.8: Theorem \ref{['thm:davclassification']}
  • Conjecture 1.9: Conjecture \ref{['conj:uqsl2']}
  • Proposition 2.1
  • ...and 92 more