# Diagram automorphisms and canonical bases for quantum groups

MS Seminar (Mathematics – String Theory)

Speaker: | Toshiaki Shoji (Tongji University) |
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Title: | Diagram automorphisms and canonical bases for quantum groups |

Date (JST): | Tue, Sep 21, 2021, 13:30 – 15:00 |

Place: | Online |

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. |

Organizers: | G. Dmitrii, T. Milanov, M. Yamazaki |
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Contact: | seminar-organizer _at_ ipmu.jp |