Given a modular category C we study the group Aut(C) of its braided autoequivalences. This group is known to be isomorphic to the group of invertible C-module categories (the Picard group of C). As a consequence, if G is a finite group then G-orbifold extensions of C can be completely classified in terms of actions G --> Aut(C).
In the simplest situation, when C is pointed, Aut(C) is the orthogonal group of a quadratic form. This fact provides a useful geometric intuition for the general case. In particular, when C is the Drinfeld double of a fusion category there is a natural action of Aut(C) on the set of Lagrangian algebras in C (i.e., on the categorical analogue of Lagrangian Grassmannian). We analyze this action and compute Aut(C) for several classes of modular categories. It turns out that the corresponding actions can often be identified with classical actions of linear groups on projective spaces.
We also discuss a non-semisimple version of these results and compute
Aut(C) when C is the Drinfeld double of a finite supergroup. This
talk is based
on joint works with Costel Bontea and Brianna Riepel.
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