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# Homology, Homotopy and Applications

## Volume 20 (2018)

### Number 1

### Third cohomology and fusion categories

Pages: 275 – 302

#### Authors

#### 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. Our computations also allow us to describe associators of Lagrangian algebra in pointed braided fusion categories.

#### Keywords

group cohomology, fusion category, finite group theory

#### 2010 Mathematics Subject Classification

18D10, 18G15

Received 3 May 2017

Published 21 February 2018