Asymmetric Twin Representation: the Transfer Matrix Symmetry

TL;DR

This paper generalizes the symmetry analysis of the asymmetric twin model's transfer matrix, establishing boundary quantum algebra as a symmetry and exploring its relation to the boundary Temperley-Lieb algebra.

Contribution

It proves the boundary quantum algebra symmetry for open transfer matrices and analyzes the symmetry reduction with non-diagonal boundaries in the asymmetric twin representation.

Findings

Boundary quantum algebra is a symmetry for open transfer matrices with trivial boundary.

Transfer matrix exhibits ${\cal U}_{q}(sl_2) \otimes {\cal U}_{{\mathrm i}}(sl_2)$ symmetry under trivial boundary conditions.

Certain boundary non-local charges commute with the transfer matrix even with non-diagonal boundaries.

Abstract

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 ${\cal U}_{q}(sl_2) \otimes {\cal U}_{{\mathrm i}}(sl_2)$ symmetry. When a non-diagonal 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.