Abstract

Lusztig has introduced semisimple perverse sheaves for quivers and the induction and restriction functors to categorify the positive part of the quantum groups and provoided the existence of the canonical basis. Even though one can use an algebraic construction to obtain the canonical basis of irreducible integrable highest weight modules, how to realize the integrable highest weight modules and their canonical bases via Lusztig’s sheaves is still an important problem. We generalize Lusztig’s theory to N-framed quivers and define certain localizations of Lusztig’s perverse sheaves to realize (tensor products of) irreducible integrable highest weight modules. As a byproduct, we give a proof of the Yang-Baxter equation by using the coassociativity of Lusztig’s restriction functor. This is a joint work with Jiepeng Fang and Jie Xiao.