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Boundary transfer matrices arising from quantum symmetric pairs

Andrea Appel, Bart Vlaar

TL;DR

We address universal boundary transfer matrices for quantum groups and quantum symmetric pairs, providing a framework that generalizes Sklyanin's two-row transfer matrices to cylindrical bialgebras and coideal subalgebras. The method uses cylindrical structures (without explicit TeX) and tensor K-matrices to define a ring homomorphism from the module Grothendieck group via a trace, with a boundary gauge transformation ensuring multiplicativity. In balanced Hopf algebras, generalized dual K-matrices yield nontrivial boundary transfers, while in the affine quantum group setting the construction produces spectral transfer matrices valued in the boundary algebra that specialize to Kolb's finite-type map; this provides a boundary analogue of q-characters and connects to two-boundary qKZ equations. The work thus unifies quantum symmetric pairs, universal R/K-matrices, and boundary transfer phenomena, offering a representation-theoretic route to boundary integrable systems and potential applications to boundary Bethe-type subalgebras and spectral theory.

Abstract

We introduce a universal framework for boundary transfer matrices, inspired by Sklyanin's two-row transfer matrix approach for quantum integrable systems with boundary conditions. The main examples arise from quantum symmetric pairs of finite and affine type. As a special case we recover a construction by Kolb in finite type. We review recent work on universal solutions to the reflection equation and highlight several open problems in this field.

Boundary transfer matrices arising from quantum symmetric pairs

TL;DR

We address universal boundary transfer matrices for quantum groups and quantum symmetric pairs, providing a framework that generalizes Sklyanin's two-row transfer matrices to cylindrical bialgebras and coideal subalgebras. The method uses cylindrical structures (without explicit TeX) and tensor K-matrices to define a ring homomorphism from the module Grothendieck group via a trace, with a boundary gauge transformation ensuring multiplicativity. In balanced Hopf algebras, generalized dual K-matrices yield nontrivial boundary transfers, while in the affine quantum group setting the construction produces spectral transfer matrices valued in the boundary algebra that specialize to Kolb's finite-type map; this provides a boundary analogue of q-characters and connects to two-boundary qKZ equations. The work thus unifies quantum symmetric pairs, universal R/K-matrices, and boundary transfer phenomena, offering a representation-theoretic route to boundary integrable systems and potential applications to boundary Bethe-type subalgebras and spectral theory.

Abstract

We introduce a universal framework for boundary transfer matrices, inspired by Sklyanin's two-row transfer matrix approach for quantum integrable systems with boundary conditions. The main examples arise from quantum symmetric pairs of finite and affine type. As a special case we recover a construction by Kolb in finite type. We review recent work on universal solutions to the reflection equation and highlight several open problems in this field.

Paper Structure

This paper contains 46 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Quantum symmetric pairs
  3. Universal K-matrices
  4. Present work
  5. Notation
  6. Acknowledgments
  7. Boundary transfer matrices for cylindrical bialgebras
  8. Quasitriangular bialgebras and twist pairs
  9. Cylindrical structures AV22
  10. Tensor K-matrices
  11. Gauge transformations and invariants
  12. Boundary gauge transformation and transfer matrices
  13. Boundary transfer matrices for balanced Hopf algebras
  14. Balanced and sovereign Hopf algebras
  15. R-matrices and dual modules
  16. ...and 31 more sections

Key Result

Theorem 2.5.2

Let $A$ be a quasitriangular bialgebra, $B \subseteq A$ a right coideal subalgebra, $(\psi,J,K)$ a cylindrical structure on $(A,B)$, and $g\in A^\times$ a boundary gauge transformation for $(\psi,J)$. Set $\overline{\mathbb{K}}_{}=(1\otimes g)\cdot{{\mathbb{K}}}_{}$. Then, there is a ring homomorphi where $V\in{\operatorname{Mod}}_{\sf fd}(A)$. In particular, $[\mathscr{T}^{(V)}, \mathscr{T}^{(W)}

Theorems & Definitions (47)

  • Example 2.1.1
  • Definition 2.2.1
  • Remark 2.2.2
  • Example 2.2.3
  • Remark 2.3.1
  • Definition 2.5.1
  • Theorem 2.5.2
  • Remark 2.5.3
  • proof : Proof of Theorem \ref{['thm:t:Grothendieck']}
  • Remark 3.1.1
  • ...and 37 more