Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 009, 19 pages      math-ph/0606040      http://dx.doi.org/10.3842/SIGMA.2007.009
Contribution to the Vadim Kuznetsov Memorial Issue

Asymmetric Twin Representation: the Transfer Matrix Symmetry

Anastasia Doikou
INFN Section of Bologna, Physics Department, University of Bologna, Via Irnerio 46, 40126 Bologna, Italy

Received August 02, 2006, in final form December 26, 2006; Published online January 09, 2007

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 Uq(sl2) Ui(sl2) 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.

Key words: quantum integrability; boundary symmetries; quantum algebras; Hecke algebras.

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