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

Kang Lu, Yung-Ning Peng, Lukas Tappeiner, Lewis Topley, Weiqiang Wang

TL;DR

This work constructs parabolic (Drinfeld-type) presentations for twisted Yangians of AI and AII, and extends these to shifted and truncated versions, establishing PBW bases and explicit relations. It then connects the truncated shifted twisted Yangians to finite $W$-algebras via universal equivariant quantizations of Slodowy varieties, showing isomorphisms in types B and C and for two-row partitions in type D, modulo a center-related conjecture in the D-case. Central to the approach are Gauss-decomposition-based parabolic generators, a new baby comultiplication, and a Dirac-reduction framework that links quantum algebras to Slodowy slices. The results provide a unified, structure-preserving realization of finite $W$-algebras as quotients of shifted twisted Yangians, enabling explicit presentations in a broad range of classical types and laying groundwork for remaining D-type cases. The work also introduces canonical filtrations, universal quantizations, and a combinatorial pyramid formalism to organize and study these objects systematically.

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}.

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

TL;DR

This work constructs parabolic (Drinfeld-type) presentations for twisted Yangians of AI and AII, and extends these to shifted and truncated versions, establishing PBW bases and explicit relations. It then connects the truncated shifted twisted Yangians to finite -algebras via universal equivariant quantizations of Slodowy varieties, showing isomorphisms in types B and C and for two-row partitions in type D, modulo a center-related conjecture in the D-case. Central to the approach are Gauss-decomposition-based parabolic generators, a new baby comultiplication, and a Dirac-reduction framework that links quantum algebras to Slodowy slices. The results provide a unified, structure-preserving realization of finite -algebras as quotients of shifted twisted Yangians, enabling explicit presentations in a broad range of classical types and laying groundwork for remaining D-type cases. The work also introduces canonical filtrations, universal quantizations, and a combinatorial pyramid formalism to organize and study these objects systematically.

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 -algebra which quantizes a suitable Slodowy slice. This provides a presentation of the finite -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}.
Paper Structure (75 sections, 94 theorems, 410 equations)

This paper contains 75 sections, 94 theorems, 410 equations.

Table of Contents

  1. Introduction
  2. History
  3. Parabolic presentations
  4. Shifted twisted Yangians
  5. Truncation, baby comultiplication and the center
  6. Dirac reduction of shifted Yangians and Slodowy slices
  7. Equivariant quantizations and finite $W$-algebras
  8. Organization
  9. Parabolic presentations of twisted Yangians
  10. Preliminaries
  11. Yangians $\mathrm{Y}(\mathfrak{gl}_N)$
  12. Twisted Yangians
  13. Homomorphisms of (extended) twisted Yangians
  14. Sklyanin determinants and minors
  15. Gauss decomposition and parabolic generators for type AI
  16. ...and 60 more sections

Key Result

Theorem A

Let $\mu = (\mu_1,...,\mu_n)$ be a strict composition of $N$, with additional assumption that all $\mu_i$ are even in type AII. Then ${\mathscr{Y}}_N^\pm$ is generated by the elements subject to either the relations pr1--pr-2 or the relations pr1--pr7 and pr-1-2-better.

Theorems & Definitions (187)

  • Theorem A: Parabolic presentations, Theorem \ref{['mainthm']}, Theorem \ref{['redthm']}
  • Theorem B: PBW basis for shifted twisted Yangians, Theorem \ref{['pbwstw']}, Theorem \ref{['a2shPBWgauss']}
  • Theorem C: Baby comultiplication, Theorem \ref{['delR']}
  • Theorem D: Truncated PBW theorem, Corollaries \ref{['loopgradedcentralizerAI']}, \ref{['loopgradedcentralizerAII']} and Theorem \ref{['truncatedcenter']}
  • Theorem E: Corollaries \ref{['gryangsliceAI']} and \ref{['gryangsliceAII']}
  • Theorem F: Theorem \ref{['t:finalisothm']}
  • Proposition 2.1: BK05, MNO96
  • Definition 2.2
  • Lemma 2.3: cf. LWZ23
  • proof
  • ...and 177 more