Modular invariants for group-theoretical modular data. I

Abstract

We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) grouptheoretical 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.

On Lagrangian algebras in group-theoretical braided fusion categories

“…This paper extends the results of [3] to the case of Z(G, α) with a nontrivial cocycle α. It could have been titled "Modular invariants for group-theoretical modular data II".…”

“…It could have been titled "Modular invariants for group-theoretical modular data II". The scheme of the proof we follow here is very similar to [3]. However, the present paper is not a mere extension of [3]: the presence of a nontrivial cocycle makes all constructions and computations much more elaborate.…”

“…The scheme of the proof we follow here is very similar to [3]. However, the present paper is not a mere extension of [3]: the presence of a nontrivial cocycle makes all constructions and computations much more elaborate. For example, to identify categories of local modules in section 2.3, we design a recognition tool by looking at more general fusion categories of group theoretic origin (the appendix).…”

The Quantum Double Model with Boundary: Condensations and Symmetries

“…6 Using this idea we show a non-trivial symmetry in Z(F + q F × q ). Let F q be the finite field with q elements and denote its additive and multiplicative groups by F + q and F × q respectively.…”

“…That is why the excitation X Y of H G×G ,U corresponds to the pair of anyons (X, Y op ) in the unfolded plane, where if Y = (y, π) then Y op = (y, π * ) (see [6] for the definition of the opposite category). Therefore, in the unfolded plane, anyons on the right hand side indeed live in the category Z(G)…”