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arXiv:1103.5526 · Journal of Physics Conference Series

Free field approach to diagonalization of boundary transfer matrix : recent advances

Takeo Kojima

Published Mar 1, 2011• 5 citations

Abstract

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 This paper includes new unpublished formula of the eigenvector for U q ( A (2) 2 ).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 ).

DOI: 10.1088/1742-6596/284/1/012041

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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 This paper includes new unpublished formula of the eigenvector for U q ( A (2) 2 ).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 ).

Infinitely many commuting operators for the elliptic quantum group

U_{q,p}(\hat{sl_N})

We construct two classes of infinitely many commuting operators associated with the elliptic quantum group

U_{q,p}(\hat{sl_N})

. We call one of them the integral of motion

{\cal G}_m

,

(m \in {\mathbb N})

and the other the boundary transfer matrix

T_B(z)

,

(z \in {\mathbb C})

. The integral of motion

{\cal G}_m

is related to elliptic deformation of the

N

-th KdV theory. The boundary transfer matrix

T_B(z)

is related to the boundary

U_{q,p}(\hat{sl_N})

face model. We diagonalize the boundary transfer matrix

T_B(z)

by using the free field realization of the elliptic quantum group, however diagonalization of the integral of motion

{\cal G}_m

is open problem even for the simplest case

U_{q,p}(\hat{sl_2})

.

We present a systematic way of constructing boundary quantum group generators associated to non-diagonal solutions of the reflection equation for the XXZ model. This is achieved by studying the asymptotics of the corresponding open spin chain. Linear intertwining relations for the

{\cal K}

matrix and Sklyanin's operator are also obtained, and the symmetry of the transfer matrix is investigated.