Modular invariants for group-theoretical modular data. I
Alexei Davydov
TL;DR
This work provides a concrete classification of indecomposable commutative separable (special Frobenius) algebras and their local modules in untwisted group-theoretical modular categories, tying these algebras to modular invariants in group-theoretical modular data. The full centre construction links Morita equivalence classes of algebras to trivialising algebras in the product category and yields a practical description of invariants via explicit characters, including a closed-form formula for the character of a trivialising algebra in terms of a subgroup $H$ and a 2-cocycle $\gamma$. The paper develops a detailed transfer framework for algebras in $\mathcal{Z}(G)$ from subgroups $H$, classifies the possible trivialising data as $(H,\gamma)$ (with generalization to products), and provides an explicit treatment of the $S_3$ case as a worked example. Overall, these results connect the algebraic classification in modular categories to computable modular invariants and to ribbon-equivalence classifications of group-theoretical modular categories, with implications for related 3D TQFTs and orbifold constructions.
Abstract
We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories.