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

TL;DR

This paper reviews the free field approach to diagonalizing boundary transfer matrices related to quantum groups, providing new formulas for eigenvectors and discussing potential extensions to more general quantum groups.

Contribution

It introduces new eigenvector formulas for boundary transfer matrices and discusses extending the diagonalization method to broader quantum groups.

Findings

Constructed free field realizations of eigenvectors.

Presented new unpublished eigenvector formulas for $U_q(A_2^{(2)})$.

Suggested extension of the method to general quantum groups.

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 $U_{q,p}(\hat{sl_N})$. We construct the free field realizations of the eigenvectors of the boundary transfer matrix $T_B(z)$. This paper includes new unpublished formula of the eigenvector for $U_q(A_2^{(2)})$.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 $U_{q,p}(g)$.