- 表示论专题

The positive part of $U_q \left( \widehat{\mathfrak{sl}_2} \right)$ and its embedding into a q-shuffle algebra

组织者

沙米尔·沙基洛夫

演讲者

阮宸炜

时间

2024年10月18日 10:00 至 11:30

地点

A3-4-101

线上

Zoom 638 227 8222 (BIMSA)

摘要

The q-deformed enveloping algebra $U_q \left( \widehat{\mathfrak{sl}_2} \right)$ and its positive part $U_q^+$ are studied in both mathematics and mathematical physics. ln 1995 , Rosso obtained an embedding of the algebra $U_q^+$ into a q-shuffle algebra. In the literature, there are three PBW bases for $U_q^+$ due to Damiani, Beck, Terwilliger respectively. The Damiani and the Beck PBW bases were originally defined using recurrence relations. Later, Terwilliger used the Rosso embedding to obtain closed forms for these two bases. Terwilliger also obtained the alternating PBW basis for $U_q^+$ using the Rosso embedding. The three PBW bases are related via exponential formulas. ln recent years, I used the Rosso embedding to obtain a uniform approach to the PBW bases and the exponential forrnulas aforementioned. I also studied sonme elements of interest related to the alternating PBW basis. We will discuss these results in this talk.
This talk is based on arXiv:2305.11152, arXiv:2408.02633

演讲者介绍

Chenwei Ruan completed his Ph.D. at the University of Wisconsin-Madison in 2024 under Paul Terwilliger. He is currently a Chern Instructor at the Beijing Institute of Mathematical Sciences and Applications. His research focuses on quantum algebras and representation theory, investigating the combinatorial structures underlying integrable systems. His recent work examines special bases and generating functions in quantum group theory, bridging abstract algebra with applications in mathematical physics.