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Twisted q-Yangians and Sklyanin determinants

Naihuan Jing, Jian Zhang

TL;DR

The paper develops a comprehensive framework for twisted $q$-Yangians as coideal subalgebras of $ ext{U}_q( rak{gl}_N)$ and extends this to the quantum affine setting, establishing a central Sklyanin determinant calculus for orthogonal and symplectic types. It introduces and analyzes determinant identities (Jacobi, Sylvester, Cayley, Muir, MacMahon) for quantum and Sklyanin determinants, and derives Liouville-type formulas connecting central elements to determinant ratios. The work provides explicit affine constructions via embeddings $S(u) o L^-(uq^{-c})L^+(u^{-1})^{t}$ (orthogonal) and $S(u) o L^-(uq^{-c})G L^+(u^{-1})^{t}$ (symplectic), together with a detailed treatment of minors, comatrices, and central series, thereby extending invariant theory to quantum affine symmetric spaces. Overall, these results yield centralized structures and determinant identities that generalize classical linear algebra to the quantum affine twisted context, with potential applications to representation theory and quantum symmetric spaces.

Abstract

$q$-Yangians can be viewed both as quantum deformations of the loop algebras of upper triangular Lie algebras and deformations of the Yangian algebras. In this paper, we study the quantum affine algebra as a product of two copies of the $q$-Yangian algebras. This viewpoint enables us to investigate the invariant theory of quantum affine algebras and their twisted versions. We introduce the twisted Sklyanin determinant for twisted quantum affine algebras and establish various identities for the Sklyanin determinants.

Twisted q-Yangians and Sklyanin determinants

TL;DR

The paper develops a comprehensive framework for twisted -Yangians as coideal subalgebras of and extends this to the quantum affine setting, establishing a central Sklyanin determinant calculus for orthogonal and symplectic types. It introduces and analyzes determinant identities (Jacobi, Sylvester, Cayley, Muir, MacMahon) for quantum and Sklyanin determinants, and derives Liouville-type formulas connecting central elements to determinant ratios. The work provides explicit affine constructions via embeddings (orthogonal) and (symplectic), together with a detailed treatment of minors, comatrices, and central series, thereby extending invariant theory to quantum affine symmetric spaces. Overall, these results yield centralized structures and determinant identities that generalize classical linear algebra to the quantum affine twisted context, with potential applications to representation theory and quantum symmetric spaces.

Abstract

-Yangians can be viewed both as quantum deformations of the loop algebras of upper triangular Lie algebras and deformations of the Yangian algebras. In this paper, we study the quantum affine algebra as a product of two copies of the -Yangian algebras. This viewpoint enables us to investigate the invariant theory of quantum affine algebras and their twisted versions. We introduce the twisted Sklyanin determinant for twisted quantum affine algebras and establish various identities for the Sklyanin determinants.
Paper Structure (15 sections, 34 theorems, 261 equations)

This paper contains 15 sections, 34 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Coideal subalgebras of $\mathrm{U}_q(\mathfrak{gl}_N)$
  3. Orthogonal case
  4. Symplectic case
  5. Quantum affine algebra $\mathrm{U}_q(\widehat{\mathfrak{gl}}_N)$
  6. Quantum (affine) determinants
  7. Minor identities for quantum determinants
  8. Liouville formula
  9. Coideal subalgebras of $\mathrm{U}_q(\widehat{\mathfrak{gl}}_N)$
  10. Orthogonal twisted $q$-Yangians
  11. Symplectic twisted -Yangians
  12. Sklyanin determinant
  13. Minor identities for Sklyanin determinants
  14. Sylvester's theorem for the Sklyanin determinant
  15. Liouville formula

Key Result

Theorem 2.1

MRS The map $S\mapsto L^-(L^+)^{t}$ defines an algebra embedding of $\mathrm{U}_{q}^{\mathrm{tw}}(\mathfrak{o}_{N}) \longrightarrow \mathrm{U}_{q} (\mathfrak{gl}_{N})$. The monomials form a basis of the algebra $\mathrm{U}_{q}^{\mathrm{tw}}(\mathfrak{o}_{N})$, where $k_{ij}$ are nonnegative integers.

Theorems & Definitions (58)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 48 more