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On extended Frobenius structures

Agustina Czenky, Jacob Kesten, Abiel Quinonez, Chelsea Walton

TL;DR

This work develops a systematic treatment of extended Frobenius structures across algebraic and categorical settings. It introduces extended Frobenius algebras over a field, catalogs their classifications for key Frobenius algebras, and then situates these structures inside monoidal categories, establishing ExtFrobAlg(\mathcal{C}) as a monoidal category and connecting to separable algebras and Hopf theory. It further defines extended Hopf algebras and shows how integral Hopf algebras yield Frobenius structures that admit extended data, with a functorial bridge to extended Frobenius algebras. The theory culminates in the notion of extended Frobenius monoidal functors, proving their basic properties, composition closure, and the formation of a 2-category ExtFrobMon, along with illustrative examples. Together, these results provide algebraic and categorical tools for modeling unoriented 2D TQFTs and related invariants via extended Frobenius data.

Abstract

A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. In 2006, Turaev and Turner introduced additional structure on Frobenius algebras, forming what are called extended Frobenius algebras, to classify 2-TQFTs in the unoriented case. This work provides a systematic study of extended Frobenius algebras in various settings: over a field, in a monoidal category, and in the framework of monoidal functors. Numerous examples, classification results, and general constructions of extended Frobenius algebras are established.

On extended Frobenius structures

TL;DR

This work develops a systematic treatment of extended Frobenius structures across algebraic and categorical settings. It introduces extended Frobenius algebras over a field, catalogs their classifications for key Frobenius algebras, and then situates these structures inside monoidal categories, establishing ExtFrobAlg(\mathcal{C}) as a monoidal category and connecting to separable algebras and Hopf theory. It further defines extended Hopf algebras and shows how integral Hopf algebras yield Frobenius structures that admit extended data, with a functorial bridge to extended Frobenius algebras. The theory culminates in the notion of extended Frobenius monoidal functors, proving their basic properties, composition closure, and the formation of a 2-category ExtFrobMon, along with illustrative examples. Together, these results provide algebraic and categorical tools for modeling unoriented 2D TQFTs and related invariants via extended Frobenius data.

Abstract

A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. In 2006, Turaev and Turner introduced additional structure on Frobenius algebras, forming what are called extended Frobenius algebras, to classify 2-TQFTs in the unoriented case. This work provides a systematic study of extended Frobenius algebras in various settings: over a field, in a monoidal category, and in the framework of monoidal functors. Numerous examples, classification results, and general constructions of extended Frobenius algebras are established.

Paper Structure

This paper contains 23 sections, 30 theorems, 39 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Extended Frobenius algebras over a field
  3. Preliminary results and examples
  4. Classification results
  5. Extended Frobenius algebras in a monoidal category
  6. Background material
  7. Monoidal categories
  8. Algebraic structures in monoidal categories
  9. Structure results
  10. Ties to separable algebras and Hopf algebras
  11. Tie to separable algebras
  12. Tie to Hopf algebras
  13. Extended Hopf algebras
  14. Extended Frobenius monoidal functors
  15. Background on monoidal functors
  16. ...and 8 more sections

Key Result

Theorem 1.2

Take $n \geq 2$, and $\omega_n \in \Bbbk$ an $n$-th root of unity. The extended structures for the Frobenius algebras below are classified, recapped as follows.

Figures (31)

  • Figure 1: Some axioms for a symmetric monoidal category.
  • Figure 2: Structure morphisms for an extended Frobenius algebra in $\cal{C}$.
  • Figure 3: Axioms for an extended Frobenius algebra in $\cal{C}$.
  • Figure 4: Proof of Lemma \ref{['lem:EFrobid']}.
  • Figure 5: Extended Frobenius structure on $B$.
  • ...and 26 more figures

Theorems & Definitions (75)

  • Definition 1.1
  • Theorem 1.2: Propositions \ref{['prop:k-class']}--\ref{['prop:xn-class']}, \ref{['prop:C2-class']}--\ref{['prop:C4-class']}, \ref{['prop:V4-class']}--\ref{['prop:T2-class']}
  • Proposition 1.3: Proposition \ref{['prop:sepExt']}
  • Proposition 1.4: Proposition \ref{['prop:eHopf']}
  • Theorem 1.5: Propositions \ref{['prop:sepisEFM']}, \ref{['prop:eFMpres']}, Theorem \ref{['thm:eFMcomp']}, Remark \ref{['rem:2-cat']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 2.3
  • ...and 65 more