Crystals and cacti in representation theory

Iva Halacheva (Northeastern University)

Abstract

One approach to studying the representation theory of Lie algebras and their associated quantum groups is through combinatorial shadows known as crystals. While the original representations carry an action of the braid group, their crystals carry an action of a closely related group known as the cactus group. I will describe how we can realize this combinatorial action both geometrically, as a monodromy action coming from a family of 'shift of argument' algebras, as well as categorically through the structure of certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint works with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.