On the Uq[sl(2)]{\cal{U}}_{q}[sl(2)] Temperley-Lieb reflection matrices

This work concerns the boundary integrability of the spin-s ${\cal{U}}_{q}[sl(2)]$ Temperley-Lieb model. A systematic computation method is used to constructed the solutions of the boundary Yang-Baxter equations. For $s$ half-integer, a general $2s(s+1)+3/2$ free parameter solution is presented. It turns that for $s$ integer, the general solution has $2s(s+1)+1$ free parameters. Moreover, some particular solutions are discussed.Comment: LaTex 17 page

Cn(1)C_{n}^{(1)}, Dn(1)D_{n}^{(1)} and A2n−1(2)A_{2n-1}^{(2)} reflection K-matrices

We investigate the possible regular solutions of the boundary Yang-Baxter equation for the vertex models associated with the $C_{n}^{(1)}$, $D_{n}^{(1)}$ and $A_{2n-1}^{(2)}$ affine Lie algebras. We find three types of solutions with $n$, $n-1$ and 1 free parameters,respectively. Special cases and all diagonal solutions are presented separately.Comment: 22 pages, LaTe

On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary

Motivated by a study of the crossing symmetry of the `gemini' representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang--Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley--Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the Boundary Yang--Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin's commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary -- despite differing from this model in having reference states.) Using this multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.Comment: 43 pages, multiple figure