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A Drinfeld type presentation of twisted Yangians

Kang Lu, Weiqiang Wang, Weinan Zhang

TL;DR

This work resolves the longstanding problem of a Drinfeld-type current presentation for twisted Yangians of type AI by employing a Gauss decomposition and quasi-determinant framework to define explicit current generators. It establishes rank-1 and rank-2 current relations, including a Serre-type relation, and proves a complete Drinfeld-type presentation for ${\mathscr{Y}}_N$ and the special twisted Yangian ${\mathscr{SY}}_N$, with a flat deformation parameterized by $\hbar$. The approach yields a unified presentation that supports generalizations to ADE split types and connects to degeneration from affine $\imath$quantum groups, shifted twisted Yangians, and finite W-algebras, enriching both representation theory and boundary integrable systems. The results provide new tools for constructing and analyzing representations, maximal commutative subalgebras, and potential parallels with parabolic and shifted twisted constructions across twisted Yangians.

Abstract

We develop a Gauss decomposition approach to establish a Drinfeld type current presentation for Olshanski's twisted Yangians associated to the orthogonal Lie algebras (also called twisted Yangians of type AI), settling a longstanding open problem. We expect that this will open the door for finding current presentations for other twisted Yangians.

A Drinfeld type presentation of twisted Yangians

TL;DR

This work resolves the longstanding problem of a Drinfeld-type current presentation for twisted Yangians of type AI by employing a Gauss decomposition and quasi-determinant framework to define explicit current generators. It establishes rank-1 and rank-2 current relations, including a Serre-type relation, and proves a complete Drinfeld-type presentation for and the special twisted Yangian , with a flat deformation parameterized by . The approach yields a unified presentation that supports generalizations to ADE split types and connects to degeneration from affine quantum groups, shifted twisted Yangians, and finite W-algebras, enriching both representation theory and boundary integrable systems. The results provide new tools for constructing and analyzing representations, maximal commutative subalgebras, and potential parallels with parabolic and shifted twisted constructions across twisted Yangians.

Abstract

We develop a Gauss decomposition approach to establish a Drinfeld type current presentation for Olshanski's twisted Yangians associated to the orthogonal Lie algebras (also called twisted Yangians of type AI), settling a longstanding open problem. We expect that this will open the door for finding current presentations for other twisted Yangians.
Paper Structure (22 sections, 28 theorems, 177 equations)

This paper contains 22 sections, 28 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. Our approach
  4. Future directions
  5. Organization
  6. Twisted Yangians of type AI
  7. Yangians
  8. Twisted Yangians
  9. Some basic properties
  10. Gauss decomposition
  11. Quasi-determinants and Gauss decomposition
  12. Properties of new current generators
  13. Special twisted Yangians
  14. Current relations for twisted Yangians of low ranks
  15. The rank 1 case
  16. ...and 7 more sections

Key Result

Lemma 2.2

Suppose $s_{ij}(u)$, $1\leqslant i,j\leqslant N$, satisfy the quaternary relation quater u. Then the relation s11 is equivalent to the following symmetry relations for all $1\leqslant i,j\leqslant N$.

Theorems & Definitions (58)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 3.1: Mol07
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • ...and 48 more