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Baxterization for the dynamical Yang-Baxter equation

Muze Ren

Abstract

The Baxterization process for the dynamical Yang-Baxter equation is studied. We introduce the local dynamical Hecke ,Temperley-Lieb and Birman-Murakami-Wenzl operators, then by inserting spectral parameters, from each representation of these operators, we get dynamical R matrix under some conditions. As applications, we reformulate trigonometric degeneration of elliptic quantum group representations and also get dynamical R matrix for critical ADE integrable lattice models. Through Baxterization, we construct some one dimensional integrable systems that are dynamical version of the Heisenberg spin chain.

Baxterization for the dynamical Yang-Baxter equation

Abstract

The Baxterization process for the dynamical Yang-Baxter equation is studied. We introduce the local dynamical Hecke ,Temperley-Lieb and Birman-Murakami-Wenzl operators, then by inserting spectral parameters, from each representation of these operators, we get dynamical R matrix under some conditions. As applications, we reformulate trigonometric degeneration of elliptic quantum group representations and also get dynamical R matrix for critical ADE integrable lattice models. Through Baxterization, we construct some one dimensional integrable systems that are dynamical version of the Heisenberg spin chain.
Paper Structure (11 sections, 20 theorems, 81 equations, 1 figure, 2 tables)

This paper contains 11 sections, 20 theorems, 81 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Main constructions
  4. Yang-Baxter operator and other local operators
  5. Local dynamical operators
  6. Relation between the usual and dynamical operators
  7. Baxterization
  8. Example: unrestricted cases
  9. Example: restricted case
  10. Transfer matrix and hamiltonian of spin chain
  11. Perron-Frobenius theorem

Key Result

Lemma 2.11

Suppose that we have dynamical Hecke operator $S(a)$ with parameter $q$, if $\bar{q}(a)+\bar{q}^{-1}(a)\ne 0$ for any $a$ in $\rm{Ob}(\pi)$, then we can define $T(a):=\bar{q}(a)-S(a)$, which forms the local dynamical Temperley--Lieb operator with parameter $\bar{\kappa}=\bar{q}+\bar{q}^{-1}$.

Figures (1)

  • Figure 1: unrestricted groupoid of type $A$

Theorems & Definitions (50)

  • Example 2.1
  • Example 2.2: Action groupoid
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 40 more