Shifted twisted Yangians and Slodowy slices in classical Lie algebras

TL;DR

This paper introduces shifted twisted Yangians for classical Lie algebras, explores their semiclassical limits, and connects them to Slodowy slices, providing new algebraic and geometric insights into their structure.

Contribution

It defines shifted twisted Yangians of type AI, studies their semiclassical limits, and relates them to Slodowy slices and Poisson algebras, extending previous work to new algebraic and geometric contexts.

Findings

Shifted twisted Yangians coincide with Dirac reductions of semiclassical shifted Yangians.

They admit truncations isomorphic to Slodowy slices for various nilpotent elements.

Provides Poisson presentations of Slodowy slices for all even nilpotent elements in types B, C, D.

Abstract

In this paper we introduce the shifted twisted Yangian of type {\sf AI}, following the work of Lu--Wang--Zhang, and we study their semiclassical limits, a class of Poisson algebras. We demonstrate that they coincide with the Dirac reductions of the semiclassical shifted Yangian for $\mathfrak{gl}_n$. We deduce that these shifted twisted Yangians admit truncations which are isomorphic to Slodowy slices for many non-rectangular nilpotent elements in types {\sf B}, {\sf C}, {\sf D}. As a direct consequence we obtain parabolic presentations of the semiclassical shifted twisted Yangian, analogous to those introduced by Brundan--Kleshchev for the Yangian of type {\sf A}. Finally we give Poisson presentations of Slodowy slices for all even nilpotent elements in types {\sf B}, {\sf C}, {\sf D}, generalising the recent work of the second author.