RTMP Seminar: Derived actions of groupoids of 2-Calabi-Yau categories

Seminar | October 17 | 2-3 p.m. | 748 Evans Hall

 Milen Yakimov, Louisiana State University

 Department of Mathematics

Starting with work of Seidel and Thomas, there has been a great interest in the construction of faithful actions of various classes of groups on derived categories (braid groups, fundamental groups of hyperplane arrangements, mapping class groups). We will describe a general construction of this sort in the setting of algebraic 2-Calabi-Yau triangulated categories. It is applicable to categories coming from algebraic geometry, cluster algebras and topology. To each algebraic 2-Calabi-Yau category, we associate a groupoid, defined in an intrinsic homological way, and then construct a representation of it by derived equivalences. In a certain general situation we prove that this action is faithful and that the green green groupoid is isomorphic to the Deligne groupoid of a hyperplane arrangement. This applies to the 2-Calabi-Yau categories arising from algebraic geometry. We will also illustrate this construction for categories coming from cluster algebras, where one gets categorical actions of braid groups. This is a joint work with Peter Jorgensen (Newcastle University).