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Journal of Physics A Mathematical and General

BoundaryK-matrix, elliptic Dunkl operator and quantum many-body systems

Kazuhiro Hikami

Published May 7, 1996• 17 citations
Publisher

Abstract

The infinite-dimensional representation of the boundary K-operator (the solution of the reflection equation) is considered. The trigonometric K-matrix is studied as a degenerate case of the elliptic operator. A method to construct the elliptic Dunkl operator is proposed, and the relationship with the quantum many-body problem is also discussed.

DOI
10.1088/0305-4470/29/9/025

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The infinite-dimensional representation of the boundary K-operator (the solution of the reflection equation) is considered. The trigonometric K-matrix is studied as a degenerate case of the elliptic operator. A method to construct the elliptic Dunkl operator is proposed, and the relationship with the quantum many-body problem is also discussed.
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We diagonalize infinitely many commuting operators T B ( z ). We call these operators T B ( z ) the boundary transfer matrix associated with the quantum group and the elliptic quantum group. The boundary transfer matrix is related to the solvable model with a boundary. When we diagonalize the boundary transfer matrix, we can calculate the correlation functions for the solvable model with a boundary. We review the free field approach to diagonalization of the boundary transfer matrix T B ( z ) associated with U q ( A (2) 2 ) and . We construct the free field realizations of the eigenvectors of the boundary transfer matrix TB ( z ). This paper includes new unpublished formula of the eigenvector for U q ( A (2) 2 ). It is thought that this diagonalization method can be extended to more general quantum group U q ( g ) and elliptic quantum group U q,p ( g ).
Some special solutions to the reflection equation are considered. These boundary matrices are defined on the common quantum space with the other operators in the chain. The relations with the Drinfeld twist are discussed. PACS: 02.90.+p, 03.20.+i,03.65.Fd, 11.30.-j 0 1 On a covariance property of the reflection equation Let the operator-valued function R(u) : C → Aut(V ⊗ V) be a solution to the quantum Yang-Baxter equation [1] in a finite-dimensional linear space V. Let us connect with a given matrix R(u) two associative algebras TR and UR generated by noncommutative entries of the square matrices T(u) and K(u) satisfying the fundamental commutator relation [1, 2] or the reflection equation [3, 4] R12(u − v) 1