Shifted twisted Yangians and Slodowy slices in classical Lie algebras
Lukas Tappeiner, Lewis Topley
TL;DR
This work develops a semiclassical framework connecting shifted twisted Yangians of type AI with classical finite $W$-algebras and Slodowy slices. It proves a Poisson isomorphism $R(y_n(\sigma),\tau) \cong y_n^+(\sigma)$, constructs parabolic presentations for the semiclassical shifted Yangians, and shows how Dirac reductions yield Poisson presentations of Slodowy slices for even nilpotent elements in types ${\bf B},{\bf C},{\bf D}$. The authors extend Brundan–Kleshchev-type parabolic presentations from type A to the classical types via Dirac reductions and establish semiclassical analogues of the BK homomorphism, linking shifted Yangians to finite $W$-algebras. Overall, the paper provides a robust, symmetry-driven bridge between shifted (twisted) Yangians, Dirac reductions, and Slodowy slices, with explicit Poisson presentations for a broad family of nilpotent orbits.
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.