|Speaker:||Toshiaki Shoji (Tongji University)|
|Title:||Diagram automorphisms and canonical bases for quantum groups|
|Date (JST):||Tue, Sep 21, 2021, 13:30 - 15:00|
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.