Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

Abstract

In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztig's perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztig's symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztig's symmetries on the whole quantum group.

Categorification of the internal braid group action for quantum groups I: 2-functoriality

“…At the level of 1-morphisms, such functors have already appeared at the categorical level in [14,15] and were given a geometric interpretation in [21,20,63,64]; however, to our knowledge, no information about extending these maps to 2-morphisms has appeared previously. As such, Theorem 1.1 initiates the study of Lusztig's operators at the 2-categorical level.…”