Algebra seminar | Third Cohomology and Fusion Categories, March 21

The Mathematics Department Algebra Seminar features Darren Simmons (Ohio University) discussing "Third Cohomology and Fusion Categories" on March 21 from 4-5 p.m. via Zoom.

Abstract:  It was observed recently that for a fixed finite group $G$, the set of all Drinfeld centers of $G$ twisted by $3$-cocycles form a group:  the so-called group of {\em modular extensions} (of the representation category of $G$), which is isomorphic to the third cohomology group of $G$ with coefficients in the multiplicative group of an algebraically closed field $F$ of characteristic zero. We show that for an Abelian $G$, pointed twisted Drinfeld centers 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 {\em 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. This is joint work with Alexei Davydov.