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A Q-operator for open spin chains II: boundary factorization

Alec Cooper, Bart Vlaar, Robert Weston

Abstract

One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang-Baxter equation associated to particular infinite-dimensional representations). To have such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang-Baxter equation) associated to these representations. In the case of quantum affine $\mathfrak{sl}_2$ and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

A Q-operator for open spin chains II: boundary factorization

Abstract

One of the features of Baxter's Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang-Baxter equation associated to particular infinite-dimensional representations). To have such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang-Baxter equation) associated to these representations. In the case of quantum affine and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.
Paper Structure (44 sections, 25 theorems, 188 equations)

This paper contains 44 sections, 25 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Background and overview
  3. Present work
  4. Outline
  5. Quantum affine $\mathfrak{s}\mathfrak{l}_2$ and its universal R-matrix
  6. General overview of finite-dimensional R-matrix theory
  7. Quantum affine $\mathfrak{s}\mathfrak{l}_2$
  8. Quantized Kac-Moody algebra
  9. Co-opposite Hopf algebra structure
  10. Weight modules
  11. Quasitriangularity
  12. Level-0 representations
  13. The action of ${\mathcal{R}}$ on tensor products of level-0 modules
  14. Adjusting the grading
  15. The augmented q-Onsager algebra, its twist and its universal K-matrix
  16. ...and 29 more sections

Key Result

Theorem 2.4

Consider a pair of level-0 representations $\pi^\pm: U_q(\widehat{\mathfrak{b}}^\pm) \to \mathop{\mathrm{End}}\nolimits(V^\pm)$. ThenNote that in Section sec:LandR we will use the notation $\mathcal{R}_{\pi^+\pi^-}(z)$ for a rescaled version of the action of the grading-shifted universal R-matrix. is well-defined and commutes with $(\pi^+ \otimes \pi^-)(\Delta(k_1)) = \pi^+(k_1) \otimes \pi^-(k_1)

Theorems & Definitions (54)

  • Definition 2.1: Quantum affine $\mathfrak{s}\mathfrak{l}_2$
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 44 more