Hopf algebras and Frobenius algebras in finite tensor categories

TL;DR

This paper explores the algebraic structures within braided tensor categories, focusing on Hopf and Frobenius algebras, and their roles in encoding modular group representations and additional symmetries.

Contribution

It establishes a detailed connection between Hopf algebras, Frobenius algebras, and modular group representations in finite tensor categories.

Findings

Hopf algebra H encodes significant structure of the category

Representations of SL(2,Z) arise from H in the category

Symmetric special Frobenius algebras relate to modular group symmetries

Abstract

We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf algebra H gives rise to representations of the modular group SL(2,Z) on various morphism spaces. We also explain how every symmetric special Frobenius algebra in a semisimple modular category provides additional structure related to these representations.