1. Introduction to i-quantum groups
- Date : 2023-02-07 (Tue) 09:00 ~ 11:00 AM
2023-02-13 (Mon) 09:00 ~ 11:00 AM
- Speaker : Weiqiang Wang (University of Virginia)
- Abstract :
In this minicourse, we shall introduce i-quantum groups arising from quantum symmetric pairs as a generalization of Drinfeld-Jimbo quantum groups. We will introduce i-divided powers and show how they conceptually lead to i-Serre relations and Serre presentations for (quasi-)split i-quantum groups. We will construct a new bar involution and i-canonical basis for any integrable module over a quantum group (viewed as modules over i-quantum groups); i-divided powers are examples of i-canonical basis elements on a (modified) i-quantum group.
2. Invitation to crystal bases for quantum symmetric pairs
- Date : 2023-02-15 (Wed) AM 10:00 ~ 12:00
2023-02-17 (Fri) AM 10:00 ~ 11:00
- Speaker : Hideya Watanabe (OCAMI)
- Abstract :
The theory of crystal bases for quantum symmetric pairs, i.e., $\imath$crystal bases, which is still in progress, is an $\imath$quantum group (also known as ``quantum symmetric pair coideal subalgebra'') counterpart of the theory of crystal bases.A goal of the theory of $\imath$crystal bases is to provide a way to recover much information about the structures of representations of $\imath$quantum groups from its crystal limit, just like the theory of crystal bases for quantum groups.In these three hours of lecture, we first review basic theory of canonical bases and crystal bases for quantum groups, and $\imath$canonical bases for $\imath$quantum groups. Then, we introduce a recent progress on the theory of $\imath$crystal bases of quasi-split locally finite type. As mentioned above, the theory of $\imath$crystal bases of arbitrary type is not completed yet. Toward a next step, we discuss how the already known theory of $\imath$crystal bases could be generalized to locally finite types. It would be a great pleasure for the speaker if the audience would be interested in and develop this ongoing project.