- Date : 2022-08-31 (Wed) 11:00 ~ 12:00
Place : 129-406 (SNU)
- Speaker : 김명호 (경희대학교)
- Title : Localization of the category of modules over quiver Hecke algebras
- Abstract :
For each w in the Weyl group, there is a subcategory C_w of the category of finite-dimensional modules over the quiver Hecke algebra, which categorifies the (quantum) coordinate ring of the unipotent subgroup N(w). More precisely, there exists a ring isomorphism from the coordinate ring C[N(w)] to the Grothendieck ring of C_w and it sends a cluster monomial to an isomorphism class of simple modules in C_w. It is known that the localization of the C[N(w)] via the frozen variables is isomorphic to the coordinate ring C[N_w] of the unipotent cell N_w.
We develop a localization process of a k-linear abelian monoidal category via a family of simple objects and apply it to the case of category C_w with the simple modules corresponding to frozen variables. It turns out that the Grothendieck ring of the localization of C_w is isomorphic to the coordinate ring C[N_w] of the unipotent cell and it respects cluster algebra structures. A remarkable fact is that any object in the localization of C_w admits a left dual and a right dual. This corresponds to an algebra automorphism on C[N_w], so called the (quantum) twist map.