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GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian

Robin Bartlett, Tomasz Przezdziecki, Lukas Tappeiner

TL;DR

This work constructs GKLO-style representations for twisted Yangians of type $\mathsf{AI}$ using the current Drinfeld presentation and realises truncations as quantisations of loci in quotients of the thick affine Grassmannian. The authors connect Poisson geometry of symmetric quotients $K_0\backslash\mathrm{Gr}_{\mu}$ with twisted Yangian coideals, formulating explicit truncated algebras ${}^{\mathsf{tw}}{\mathbf{Y}}_{\mu}^{\lambda}$ via twisted GKLO representations. They establish a robust bridge between the algebraic (shifted and twisted) Yangians, their RTT and Sklyanin-minor realizations, and the geometric objects arising from the Dirac-reduced Poisson structures on loop groups. A key contribution is the explicit GKLO-type homomorphism for shifted twisted Yangians, including a symmetrised (square-root) variant and an ABCD formulation, yielding concrete quantisations of symmetric slices in affine Grassmannian quotients. The results illuminate the quantum-duality picture for twisted coideals and lay groundwork for further reductions/reducedness conjectures in the twisted setting, with potential ties to i-Coulomb branches and parallel developments in related Lie types.

Abstract

We construct an analogue of Gerasimov-Kharchev-Lebedev-Oblezin (GKLO) representations for twisted Yangians of type $\mathsf{AI}$, using the recently found current presentation of these algebras due to Lu, Wang and Zhang. These new representations allow us to define interesting truncations of twisted Yangians, which, in the spirit of Ciccoli-Drinfeld-Gavarini quantum duality, reflect the Poisson geometry of homogeneous spaces. As our main result, we prove that a truncated twisted Yangian quantizes a scheme supported on quotients of transverse slices in the affine Grassmannian.

GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian

TL;DR

This work constructs GKLO-style representations for twisted Yangians of type using the current Drinfeld presentation and realises truncations as quantisations of loci in quotients of the thick affine Grassmannian. The authors connect Poisson geometry of symmetric quotients with twisted Yangian coideals, formulating explicit truncated algebras via twisted GKLO representations. They establish a robust bridge between the algebraic (shifted and twisted) Yangians, their RTT and Sklyanin-minor realizations, and the geometric objects arising from the Dirac-reduced Poisson structures on loop groups. A key contribution is the explicit GKLO-type homomorphism for shifted twisted Yangians, including a symmetrised (square-root) variant and an ABCD formulation, yielding concrete quantisations of symmetric slices in affine Grassmannian quotients. The results illuminate the quantum-duality picture for twisted coideals and lay groundwork for further reductions/reducedness conjectures in the twisted setting, with potential ties to i-Coulomb branches and parallel developments in related Lie types.

Abstract

We construct an analogue of Gerasimov-Kharchev-Lebedev-Oblezin (GKLO) representations for twisted Yangians of type , using the recently found current presentation of these algebras due to Lu, Wang and Zhang. These new representations allow us to define interesting truncations of twisted Yangians, which, in the spirit of Ciccoli-Drinfeld-Gavarini quantum duality, reflect the Poisson geometry of homogeneous spaces. As our main result, we prove that a truncated twisted Yangian quantizes a scheme supported on quotients of transverse slices in the affine Grassmannian.
Paper Structure (42 sections, 39 theorems, 146 equations)

This paper contains 42 sections, 39 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Basic setup
  4. Affine grassmannians
  5. Commutators
  6. Symmetric quotients of the affine grassmannian
  7. Poisson structures on loop groups
  8. Symmetric quotients and fixed points
  9. Symplectic leaves and their closures
  10. Ideal generators
  11. Shifted Yangians
  12. Drinfeld presentation
  13. PBW theorem
  14. Canonical filtration
  15. Quantum duality
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $K \subset G((z^{-1}))$ denote the subgroup defined by $g(-z)^t = g(z)^{-1}$ and set $K_0 = K \cap \mathcal{G}_0$. Then

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 62 more