Simple Construction of Elliptic Boundary K - matrix ¶

We give the infinite-dimensional representation for the ellipticK-operator satisfying the boundary Yang-Baxter equation. By restricting the functional space to finite-dimensional space, we construct the elliptic K-matrix associated to Belavin’s completely Z-symmetric R-matrix. PACS: ¶ arch-ive/9507123 † [email protected] 1 The quantum R-matrix as solutions of the Yang-Baxter equation (YBE) has received much attention in mathematical physics. The algebraic structure reveals as quantum group. Recently the R-matrix has been treated as operator acting on functional space. In one sense this gives the infinite-dimensional representation for solutions of YBE. By use of operator description for R-matrix, the construction of R-matrix, especially for elliptic case, becomes much simpler. Based on the elliptic R-operator defined by Shibukawa and Ueno [1], Felder and Pasquier constructed Belavin’s elliptic R-matrix [2]. The R-matrix has been used to study spin chains with periodic boundary condition in terms of the quantum inverse scattering method. Besides the R-matrix, the other matrix called K-matrix is used to solve the spin chain with open boundary [3, 4]. In this letter we propose a method to construct the boundary K-matrix associated with Belavin’sR-matrix. Throughout this paper we use the doubly periodic function σμ(z) ≡ σμ(z, τ ), σμ(z + 1) = σμ(z), σμ(z + τ ) = e 2πiμ σμ(z), where τ is an arbitrary complex number, satisfying Im τ > 0. The function σμ(z) only has simple poles on the lattice Z+ τZ, and the residue at origin equals to one. Note that the function σμ(z) can be explicitly written as σμ(z) = θ1(z − μ, τ)θ1(0, τ ) θ1(z, τ )θ1(−μ, τ) , (1) where θ1(z, τ ) is the Jacobi’s theta function, θ1(z, τ ) = ∑ n∈Z+ 2 exp ( iπnτ + 2πin ( z + 1 2 )) . (2) For the elliptic function σμ(z), we have following lemma; Lemma 1. The elliptic function σμ(z) defined in (1) satisfies the following identities; (a) σμ(z) = −σz(μ) , (b) σμ(z) = −σ−μ(−z),