Shifted twisted Yangians and Slodowy slices in classical Lie algebras

In this paper we introduce the shifted twisted Yangian of type 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 ₙ. We deduce that these shifted twisted Yangians admit truncations which are isomorphic to Slodowy slices for many non-rectangular nilpotent elements in types B, C, 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 A. Finally we give Poisson presentations of Slodowy slices for all even nilpotent elements in types B, C, D, generalising the recent work of the second author.