Shifted twisted Yangians and finite $W$-algebras of classical type

TL;DR

This paper develops new algebraic presentations for twisted Yangians and finite W-algebras of classical types, establishing isomorphisms and PBW bases, and proposes conjectures for remaining cases.

Contribution

It introduces parabolic presentations for twisted Yangians of types AI and AII, and connects truncated shifted twisted Yangians to finite W-algebras for classical types.

Findings

Established PBW bases for all (truncated) shifted twisted Yangians of types AI and AII.

Proved isomorphism between truncated twisted shifted Yangians and finite W-algebras for certain classical types.

Provided presentations for finite W-algebras associated with specific nilpotent elements.

Abstract

We introduce parabolic presentations of twisted Yangians of types AI and AII, interpolating between the R-matrix presentation and the Drinfeld presentation. Then we formulate and provide parabolic presentations for the shifted twisted Yangians. We define quotient algebras known as truncated shifted twisted Yangians and equip them with baby comultiplications, generalizing the work of Brundan and Kleshchev. PBW bases for all (truncated) shifted twisted Yangians of type AI and AII are established along the way. Applying the theory of universal equivariant quantizations of conic symplectic singularities we show that the truncated twisted shifted Yangian is isomorphic to the finite $W$-algebra which quantizes a suitable Slodowy slice. This provides a presentation of the finite $W$-algebra associated with every even nilpotent element in type {\sf B} and {\sf C}, as well as every nilpotent element with two Jordan blocks in type {\sf D}. Finally we make a conjecture which would supply presentations in the remaining even cases in type {\sf D}.