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学术报告124: 彭勇宁 — Lectures on finite W-(super)algebras and Yangians

时间:2022-11-18 作者: 点击数:

报告时间:2022年1130日,127日,1214日,1221日,1228日(周三)930-12:00

报告地点:腾讯会议307 5128 2239

报告人:  彭勇宁 教授

工作地点:成功大学 学院

举办单位:学院

报告摘要: In this lectures, we will illustrate a remarkable relationship between type A finite W-algebras and Yangians, and its generalization to type A Lie superalgebra. Let e be a nilpotent element. Associated to e, one defines an object called finite W-algebra, denoted by We. It can be regarded as some sort of refinement of U(g), the universal enveloping algebra. However, its structure is much more complicated than U(g) and hence difficult to study. On the other hand, the Yangian Yn associated to the general linear Lie algebra gln was considered in the works of mathematical physicists from St. Petersburg, where the Yangians for other types of simple Lie algebras were later defined by Drinfeld. Yn is a deformation of U(gln[x]), the universal enveloping algebra of the polynomial current Lie algebra. It shares many nice properties of the universal enveloping algebra, together with extra fascinating symmetries that can’t be observed from Lie algebra. The relationship between We and Yn was firstly observed by Ragoucy-Sorba when e is rectangular, which means that the size of Jordan blocks of e are all the same. In this case, they showed that We is isomorphic to a quotient of Yn. This strict restriction on e was later removed by Brundan-Kleshchev, who proved that for an arbitrary nilpotent e, the associated We is isomorphic to a quotient of shifted Yangian Yn(σ), which is a subalgebra of Yn associated to a shift matrix σ related to e. In particular, the result of Ragoucy-Sorba corresponds to the case when σ = 0, where the shifted Yangian is just Yn itself. This remarkable result provides a presentation of the finite W-algebra We in terms of generators and relations, which becomes a fundamental tool to the study of We and their representation theory developed in a series of their subsequent papers. The goal of this mini course is to provide some detail about this presentation of We in terms of the shifted Yangian, and to explain a generalization of this result to the case of general linear Lie superalgebras recently established by Peng. The course can be roughly divided into four parts. 1. In the first part, we will recall some basic facts of Yn, especially the RTT presentation and parabolic presentation. Then we explain how to define the shifted Yangian Yn(σ). 2. In the second part, we recall some basic facts about finite W-algebras. In particular, we will explain the combinatorial object called pyramids that provide great visualization of e and We as well. 3. In the third part, we explain how the isomorphism between We and a quotient of Yn(σ) is obtained. 1 4. In the last part, we will explain the super version of the previous results. We will explain how to define the finite W-superalgebra We, where e is now a nilpotent element in the even part of the general linear Lie superalgebra glM|N . We will recall some basic facts of Ym|n, the Yangian of glm|n associated to an arbitrary 01-sequence, and then define its shifted version Ym|n(σ). Finally we explain how to establish the isomorphism between We and a quotient of Ym|n(σ).



报告人简介: 彭勇宁,成功大学学院教授。主要研究方向为 Yangian(杨氏代数)的结构和表示理论。已在Adv.Math, Comm. Math. Phys., J. Algebra, Lett.Math. Phys.等杂志上发表论文。


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