Seminar| Institute of Mathematical Sciences
Time: Wednesday, June 03th, 2026,14:00-15:00
Location: IMS, RS408
Speaker: Toshiaki Shoji, Tongji University
Abstract: Quantum affine algebras are affine analogue of quantum groups of finite type. Let U_q^- be the negative half of the quantum affine algebras. In the finite case, there are important bases of U_q^-, called PBW basis, canonical basis and monomial basis. In the affine case, Beck-Nakajima constructed a PBW basis of U_q^-, as an analogue of the finite case, and defined the canonical basis by using them. Beck-Nakajima's PBW basis, and canonical basis play an important role in the representation theory of quantum affine algebras.
In this talk, we construct a monomial basis of U_q^- associated to Beck-Nakajima's PBW basis. The construction of PBW basis and monomial basis is much more complicated compared to the finite case. The monomial basis has a close relation with Lusztig's geometric construction of canonical basis. By using the monomial basis, we show that there exists a simple algorithm of computing the transition matrix of PBW basis and canonical basis, which is an analogue of the finite case.