Equivariant homology theory and twisted Yangian

TL;DR

This paper constructs an algebra related to equivariant homology on Steinberg varieties and shows its connection to twisted Yangians, revealing new algebraic structures and their deformations in the context of symmetric pairs.

Contribution

It introduces a new algebra $ ilde{Y}$ that maps to equivariant homology and establishes the twisted Yangian as a quotient and deformation of this algebra.

Findings

Defined algebra $ ilde{Y}$ mapping to equivariant homology

Established twisted Yangian as a quotient of $ ilde{Y}$

Proved twisted Yangian as a deformation of twisted current algebra

Abstract

We study the convolution algebra $H^{G\times \CC^{*}}_{*}(Z)$ of $G$-equivariant homology group on the Steinberg variety of type B/C and define an algebra $\widetilde{Y}$ that maps to $H^{G\times \CC^{*}}_{*}(Z)$. The Drinfeld new realization of the twisted Yangian associated to symmetric pairs is a quotient of $\widetilde{Y}$. We also study the $G$-equivariant case and prove that the twisted Yangian is the deformation of the twisted current algebra.