MS Seminar (Mathematics - String Theory)

Speaker: Toshiaki Shoji (Tongji University)
Title: Diagram automorphisms and canonical bases for quantum groups
Date (JST): Tue, Sep 21, 2021, 13:30 - 15:00
Place: Online
Related File: 2714.pdf
Abstract: Let U be the quantum group associated to a Kac-Moody algebra of symmetric type,

and U_1 the quantum group obtained from an admissible diagram automorphism s on U.

Let U^-, U_1^- be the negative part of U, U_1, respectivey. Lusztig constructed

the canonical basis B of U^- , and the canonical signed basis (B_1)' of U_1^-,

by using the geometric theory of quivers. Then he constructed the canonical basis B_1 of U_1^-

from (B_1)' by using Kashiwara's theory of crystals, and obtained the natural bijection
between the set B^s of s-fixed elements in B and B_1.

In this talk, we take a different approach for this problem. Assuming
the existence of the canonical basis B of U^-, we construct the canonical
signed basis (B_1)' of U_1^- , and the bijection between (B')^s and (B_1)',
in an elmentary way. In the case where the order of s is odd, we can construct

the canonical basis B_1, and the bijection between B^s and B_1.