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Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras

Andrea Appel, Bart Vlaar

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-module gives rise to a family of trigonometric solutions of Cherednik's generalized reflection equation. These depend upon the choice of a quantum affine symmetric pair $U_q(\mathfrak{k})\subset U_q(\hat{\mathfrak{g}})$. Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs, obtained in our previous work, as well as the fact that every irreducible $U_q(\hat{\mathfrak{g}})$-module is generically irreducible under restriction to $U_q({\mathfrak{k}})$. In the case of small modules and Kirillov-Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.

Trigonometric K-matrices for finite-dimensional representations of quantum affine algebras

Abstract

Let be a complex simple Lie algebra and the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional -module gives rise to a family of trigonometric solutions of Cherednik's generalized reflection equation. These depend upon the choice of a quantum affine symmetric pair . Our result relies on the construction of universal K-matrices for arbitrary quantum symmetric pairs, obtained in our previous work, as well as the fact that every irreducible -module is generically irreducible under restriction to . In the case of small modules and Kirillov-Reshetikhin modules, we obtain new solutions of the standard and the transposed reflection equations.
Paper Structure (43 sections, 32 theorems, 89 equations)

This paper contains 43 sections, 32 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgments
  4. Quantum affine algebras and their R-matrices
  5. Affine Lie algebras Kac90
  6. Drinfeld-Jimbo presentation of the quantum affine algebra
  7. Category ${{\mathcal{O}}_\infty}$ and finite-dimensional modules
  8. Completions
  9. The universal R-matrix
  10. The spectral R-matrix
  11. Trigonometric R-matrices
  12. Quantum affine symmetric pairs
  13. Generalized Satake diagrams
  14. Pseudo-involutions
  15. Quantum pseudo-involutions
  16. ...and 28 more sections

Key Result

Theorem 1

Let $V$ be a finite-dimensional $U_q(\widehat{{\mathfrak g}})$-module and denote by $V^\psi$ the $\psi$-twisted module $\psi^*(V)$. The universal K-matrix $K$ gives rise to a $U_q(\mathfrak{k})$-intertwiner which satisfies the generalized reflection equation eq:CRE-intro with respect to the R-matrices ${{R}}_{V^{\psi}\, V^\psi}(z)$, ${{R}}_{{V}^{\psi}V}(z)$ and ${{R}}_{VV}(z)$.

Theorems & Definitions (68)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.3.1
  • Theorem 2.6.1
  • Theorem 2.6.1
  • Remark 2.6.2
  • Theorem 2.7.1
  • Example 2.7.3
  • Definition 3.1.1: RV20RV21
  • ...and 58 more