Autoequivalences of triangulated categories via Hochschild cohomology

Abstract: I will talk about a general tool which allows one to study symmetries of (enhanced) triangulated categories in the form of their derived Picard groups. In general, these groups are rather elusive to computations which require a rather good understanding of the whole category at hand. A result of Keller shows that the Lie algebra of the derived Picard group of an algebra can be identified with its Hochschild cohomology equipped with the Gerstenhaber Lie bracket. Mimicking the classical relationship between Lie groups and Lie algebras, I will explain how to "integrate'' elements in the Hochschild cohomology of a dg category over fields of characteristic zero to elements in the derived Picard group via a generalized exponential map. Afterwards we discuss properties of this exponential and a few applications. This includes necessary conditions for the uniqueness of enhancement of triangulated functors and uniqueness of Fourier-Mukai kernels. Other applications concern derived Picard groups of categories arising in algebra and geometry such as derived categories of graded gentle algebras and their corresponding partially wrapped Fukaya categories.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.