We construct a free field realization of the ground state of the boundary transfer matrix for the U q,p ( sl(3, C)) ABF model.Using this ground state and type-II vertex operator, we have a diagonalization of the boundary transfer matrix.
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 This paper includes new unpublished formula of the eigenvector for U q ( A (2) 2 ).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 ).
The symmetry of the Hamiltonian describing the asymmetric twin model was partially studied in earlier works, and our aim here is to generalize these results for the open transfer matrix.In this spirit we first prove, that the so called boundary quantum algebra provides a symmetry for any generic -independent of the choice of model -open transfer matrix with a trivial left boundary.In addition it is shown that the boundary quantum algebra is the centralizer of the B type Hecke algebra.We then focus on the asymmetric twin representation of the boundary Temperley-Lieb algebra.More precisely, by exploiting exchange relations dictated by the reflection equation we show that the transfer matrix with trivial boundary conditions enjoys the recognized U q (sl 2 ) ⊗ U i (sl 2 ) symmetry.When a nondiagonal boundary is implemented the symmetry as expected is reduced, however again certain familiar boundary non-local charges turn out to commute with the corresponding transfer matrix.