Third cohomology and fusion categories

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© 2018, International Press. 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.