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Quantum super-spherical pairs

D. Algethami, A. Mudrov, V. Stukopin

TL;DR

The paper develops a $ abla Z_2$-graded framework for quantum symmetric pairs by constructing quantum super-spherical pairs as coideal subalgebras inside general linear and orthosymplectic quantum supergroups $U_q( rak{g})$. It builds a systematic classification using graded Satake diagrams and formulates a $ abla Z_2$-graded Reflection Equation to produce $K$-matrices that realize quantum isotropy subgroups, linking them to left coideal subalgebras. Explicit K-matrices are derived for type $A$, $B$, and $C$ in the graded setting, with mixture parameters encoding the deformation data and root-vector constructions $F_{ ilde{oldsymbol{eta}}}$ guiding the coideal generation. The work provides a solid foundation for quantum super-symmetric spaces by connecting graded Satake data to coideal subalgebra quantization and graded RE solutions, and it outlines conjectural extensions for more intricate type II diagrams and twisted RE variants, signaling directions for future integrable-system applications.

Abstract

We introduce quantum super-spherical pairs as coideal subalgebras in general linear and orthosymplectic quantum supergroups. These subalgebras play a role of isotropy subgroups for matrices solving $\mathbb{Z}_2$-graded reflection equation. They generalize quantum (pseudo)-symmetric pairs of Letzter-Kolb-Regelskis-Vlaar.

Quantum super-spherical pairs

TL;DR

The paper develops a -graded framework for quantum symmetric pairs by constructing quantum super-spherical pairs as coideal subalgebras inside general linear and orthosymplectic quantum supergroups . It builds a systematic classification using graded Satake diagrams and formulates a -graded Reflection Equation to produce -matrices that realize quantum isotropy subgroups, linking them to left coideal subalgebras. Explicit K-matrices are derived for type , , and in the graded setting, with mixture parameters encoding the deformation data and root-vector constructions guiding the coideal generation. The work provides a solid foundation for quantum super-symmetric spaces by connecting graded Satake data to coideal subalgebra quantization and graded RE solutions, and it outlines conjectural extensions for more intricate type II diagrams and twisted RE variants, signaling directions for future integrable-system applications.

Abstract

We introduce quantum super-spherical pairs as coideal subalgebras in general linear and orthosymplectic quantum supergroups. These subalgebras play a role of isotropy subgroups for matrices solving -graded reflection equation. They generalize quantum (pseudo)-symmetric pairs of Letzter-Kolb-Regelskis-Vlaar.
Paper Structure (18 sections, 22 theorems, 143 equations)

This paper contains 18 sections, 22 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Current work
  3. Preliminaries
  4. Quantum Supergroup $U_q (\mathfrak{g})$
  5. Basic Quantum Supergroups
  6. Natural Representations
  7. Graded Reflection Equation
  8. $\mathbb{Z}_2$-graded Reflection Equation
  9. K-matrices for ortho-symplectic quantum groups
  10. Classical super-spherical pairs
  11. Weyl operator and spherical data
  12. Decorated diagrams and selection rules
  13. Graded Satake diagrams
  14. Spherical pairs and Reflection Equation
  15. Coideal subalgebras and K-matrices
  16. ...and 3 more sections

Key Result

Theorem 3.1

The following matrices satisfy the Reflection Equation associated with ortho-symplectic quantum groups: where the parameters $y_i,z_i, x_i \in \mathbb{C}$ satisfy the conditions

Theorems & Definitions (54)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • proof
  • Remark 4.4
  • Definition 4.5
  • Proposition 4.6
  • ...and 44 more