The braided monoidal structures on a class of linear Gr-categories

TL;DR

This paper classifies the braided monoidal structures on certain linear Gr-categories by explicitly computing 3-cocycles and quasi-bicharacters for finite abelian groups, enhancing understanding of their algebraic structures.

Contribution

It provides a detailed classification of braided monoidal structures on linear Gr-categories associated with finite abelian groups, using explicit cohomological computations.

Findings

Explicit classification of braided monoidal structures for groups of the form Z_m x Z_n

Computation of 3-cocycles for finite abelian groups

Identification of quasi-bicharacters relevant to the structures

Abstract

A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations of the 3-cocycles and quasi-bicharacters of finite abelian groups which are direct product of two cyclic groups.