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

We diagonalize infinitely many commuting operators TB(z). We call these operators TB(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 TB(z) associated with Uq(A(2)2) and Uq,p. 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 This paper includes new unpublished formula of the eigenvector for Uq(A(2)2). It is thought that this diagonalization method can be extended to more general quantum group Uq(g) and elliptic quantum group Uq,p(g).