Two lectures by Milen Yakimov,
chair of the group on Cluster Algebras and Quantum Groups at the ICMS, IMI-BAS.
Tuesday, March 10, at 13:00 in room 403, IMI-BAS.
Cluster structures on quantum groups and the full Berenstein-Zelevinsky conjecture
Tuesday, March 17, at 13:00 in room 403, IMI-BAS.
Reflective centres of module categories and quantum K-matrices
Abstracts:
March 10: Cluster Algebras were invented by Fomin and Zelevinsky in the early 2000s and quickly turned into an indispensable tool in many areas of mathematics. In this talk, we will provide a gentle introduction to the subject. From the early days of the theory of cluster algebras, one of the desired constructions was to turn the quantized coordinate rings of complex simple Lie groups into quantum cluster algebras. This remained elusive. We will present a resolution of this conjecture and of the Berenstein-Zelevinsky conjecture on cluster structures on quantum double Bruhat cells. This is a joint work with Fan Qin and Hironori Oya.
March 17: Braided monoidal categories have applications in various situations. In particular, their universal R-matrices give solutions of the quantum Yang-Baxter equation and representations of braid groups of type A. There are powerful methods for constructing them: Drinfeld doubles of Hopf algebras and Drinfeld centres of monoidal categories. On the other hand, universal K-matrices, leading to solutions of the reflection equation and representations of braid groups of type B are much less well understood and there are no general constructions for them. We will describe a construction of reflective centers of module categories. It gives rise to a quantum double construction for universal K-matrices, which can be viewed as a type B analog of Drinfeld’s double construction. This is a joint work with Robert Laugwitz and Chelsea Walton.
More information can be found on the seminar’s web-page: