Categorification of canonical basis and PBW basis
数学专题报告
报告题目(Title)：Categorification of canonical basis and PBW basis
报告人(Speaker)：吴雨檬（北京大学）
地点(Place)：后主楼1220
时间(Time)：2025年12月10日（周三）14:00-14:50
邀请人(Inviter)：肖杰、覃帆、周宇
报告摘要
In the presentation, we will study negative part of quantum groups following Hall algebras methods. For finite types of symmetrizable cases, we will give two ways to find the relation between PBW basis and canonical basis.
We first provide a geometric realization of the coefficients between PBW basis and the canonical basis via standard sheaves on quiver moduli spaces with admissible automorphisms. This realization is constructed through Lusztig sheaves equipped with periodic functors and their modified Grothendieck groups. Within this geometric framework, we present an alternative proof for the existence of Hall polynomials originally due to Ringel. Second, by using extension algebra arising from Lusztig's category of perverse sheaves on quiver varieties with an admissible automorphism, leading to a folding Khovanov–Lauda–Rouquier (KLR) algebras. We prove that the transition matrix between the classes of indecomposable projectives (canonical basis) and dual standard modules (PBW basis) is upper triangular with diagonal entries equal to one. In the last part, we will give some ideas of the generalization of the methods above into the Kronecker quiver case.
