Asymmetric Twin Representation: the Transfer Matrix Symmetry

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 ${\mathcal{U}}_q(s{l}_2)\otimes {\mathcal{U}}_{\mathrm{i}}(s{l}_2)$ 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.

arXiv commentThis is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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Cited by in corpus (2)

• Quantum symmetry algebras of spin systems related to Temperley-Lieb R-matrices
• Junction type representations of the Temperley-Lieb algebra and associated symmetries