# Third Cohomology and Fusion Categories

## TL;DR

This paper explores the relationship between third cohomology groups of abelian finite groups and fusion categories, specifically through the structure of twisted Drinfeld centres and Lagrangian extensions.

## Contribution

It identifies the subgroup of pointed twisted Drinfeld centres as Lagrangian extensions and computes this group, linking third cohomology to fusion category structures.

## Key findings

- Pointed twisted Drinfeld centres form a subgroup of modular extensions.
- The subgroup of Lagrangian extensions is explicitly identified and computed.
- Provides an interpretation of third cohomology in terms of fusion categories.

## Abstract

It was observed recently that for a fixed finite group $G$, the set of all Drinfeld centres of $G$ twisted by 3-cocycles form a group, the so-called group of modular extensions (of the representation category of $G$), which is isomorphic to the third cohomology group of $G$. We show that for an abelian $G$, pointed twisted Drinfeld centres of $G$ form a subgroup of the group of modular extensions. We identify this subgroup with a group of quadratic extensions containing $G$ as a Lagrangian subgroup, the so-called group of Lagrangian extensions of $G$. We compute the group of Lagrangian extensions, thereby providing an interpretation of the internal structure of the third cohomology group of an abelian $G$ in terms of fusion categories.