Categrification of Canonical Basis and PBW Basis
发布时间：2026年04月16日
浏览次数：7
发布者: He Liu
主讲人： 吴雨檬（数学中心）
活动时间： 从 2026-04-17 15:00 到 16:00
场地： 北京国际数学研究中心，镜春园78号院（怀新园）77201室
This talk discusses a geometric and categorical approach to canonical bases and PBW bases arising from Hall algebras of quivers with admissible automorphisms. We begin with the Hall algebra via function of finite-dimensional basic hereditary algebra over a finite field, and recall how the composition subalgebra of it recovers the negative part of the quantum group attached to a symmetrizable Cartan matrix. We then review Lusztig’s sheaf-theoretic construction of $\mathbb{U}^-$
via flag varieties, induction and restriction functors, and simple perverse sheaves, together with its extension to the symmetrizable setting through periodic functors induced by quiver automorphisms. The main part of the talk is devoted to recent joint work with Yixin Lan and Jie Xiao, where we study the categorification of canonical and PBW bases of symmetrizable cases in derived categories of equivariant sheaf complexes. In particular, we explain how canonical basis elements, represented by intersection cohomology complexes, can be expanded in terms of standard sheaf complexes in a modified Grothendieck group, and how the coefficients are described by explicit polynomials encoding eigenvalue data of the automorphism action on stalk cohomology. We also discuss the resulting triangularity, integrality, and polynomiality properties, which provide a sheaf-theoretic interpretation of the transition between canonical bases and PBW-type bases in the symmetrizable case.
