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A Drinfeld type presentation of twisted Yangians of quasi-split type

Kang Lu, Weinan Zhang

Abstract

We formulate a family of algebras, twisted Yangians (of simply-laced quasi-split type) in Drinfeld type current generators and defining relations. These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras. For all quasi-split type excluding the even rank case in type AIII, we show that the twisted Yangians can be realized via a degeneration on the Drinfeld type presentation of affine $\imath$quantum groups. For both even and odd rank cases in type AIII, we use the Gauss decomposition method to show that these new algebras are isomorphic to Molev-Ragoucy's reflection algebras defined in the R-matrix presentation.

A Drinfeld type presentation of twisted Yangians of quasi-split type

Abstract

We formulate a family of algebras, twisted Yangians (of simply-laced quasi-split type) in Drinfeld type current generators and defining relations. These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras. For all quasi-split type excluding the even rank case in type AIII, we show that the twisted Yangians can be realized via a degeneration on the Drinfeld type presentation of affine quantum groups. For both even and odd rank cases in type AIII, we use the Gauss decomposition method to show that these new algebras are isomorphic to Molev-Ragoucy's reflection algebras defined in the R-matrix presentation.
Paper Structure (43 sections, 63 theorems, 287 equations)

This paper contains 43 sections, 63 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Goal
  4. Drinfeld type presentation for twisted Yangians
  5. Degeneration approach
  6. Gauss decomposition approach
  7. Application
  8. Organization
  9. Twisted loop and current algebras
  10. From loop algebras to current algebras
  11. Twisted loop and current algebras
  12. Presentations of ${\mathrm{U}}(\mathfrak{g}[z]^{\check\omega})$
  13. Twisted Yangians in Drinfeld type presentations
  14. Twisted Yangians of quasi-split type
  15. PBW type bases for ${\mathbf{Y}^\imath}$
  16. ...and 28 more sections

Key Result

Lemma 2.1

There is an algebra isomorphism $\varrho: {\mathrm{U}}(\mathfrak{g}[z]) \stackrel{\cong}{\longrightarrow} \mathrm{Gr}_{\kappa}\mathfrak{g}[t,t^{-1}]$, sending $g z^r\mapsto \overline{g_{r,1}}$, for $g\in \mathfrak{g},r\in\mathbb{N}$.

Theorems & Definitions (133)

  • Lemma 2.1: KLWZ23b
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 123 more