[Seminar 2021.04.29] A realization of type A finite $W$-(super)algebra in terms of (super) Yangian

by qsms posted Apr 09, 2021


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일정시작 2021-04-29
배경색상 #77CC00

Date:  29 Apr. (Thu.) AM 10:00 ~ 10:50   (10min break)  11:00 ~ 11:50 

                                     11:00 ~ 13:00  (KST)

Place:  Zoom, Meeting ID
Title:  A realization of type A finite $W$-(super)algebra in terms of (super) Yangian

Speaker:  Yung-Ning Peng (National Central University, Taiwan)



Let $e\in\mathfrak{g}=\mathfrak{gl}_N$ be a nilpotent element. Associated to $e$, one defines an object called {\em finite $W$-algebra}, denoted by $\mathcal{W}_e$. It can be regarded as a generalization of $U(\mathfrak{g})$, the universal enveloping algebra. However, its algebraic structure is much more complicated than $U(\mathfrak{g})$ and hence difficult to study except for some special choices of $e$.


In the first part of this talk, we will explain how to obtain a realization of $\mathcal{W}_e$ in terms of the Yangian $Y_n=Y(\mathfrak{gl}_n)$ associated to $\mathfrak{gl}_n$, where $n$ denotes the number of Jordan blocks of the nilpotent $e\in\mathfrak{gl}_N$. The remarkable connection between finite $W$-algebra and Yangian was firstly observed by Ragoucy-Sorba for special choice of $e$. The general case (which means for an arbitrary $e$) was established by Brundan-Kleshchev. In particular, a certain subalgebra of $Y_n$, called the {\em shifted Yangian}, was explicitly defined. Some necessary background knowledge about finite $W$-algebra and shifted Yangian will be recalled.


In the second part of this talk, we will explain the extension of the aforementioned connection to the case of general linear Lie superalgebra. With some mild technical modifications, the finite $W$-superalgebra $\mathcal{W}_e$ can also be defined for a given nilpotent element $e\in(\mathfrak{gl}_{M|N})_{\overline{0}}$. On the other hand, the super Yangian was defined and studied by Nazarov. Therefore, it is natural to seek for a super-analogue of the aforementioned connection.

For some special choices of $e$, such a connection was established by Briot-Ragoucy for rectangular nilpotent case and Brown-Brundan-Goodwin for principal nilpotent case. However, a universal treatment for a general $e$ was still missing in the literature until our recent result. We will explain some difficulties in the Lie superalgebra setting and how to overcome them by making use of the notion of 01-sequence.







     South Korea - Seoul                                Taiwan - Taipei


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