Boundary matrices for the higher spin six vertex model

Abstract

In this paper we consider solutions to the reflection equation related to the higher spin stochastic six vertex model. The corresponding higher spin R-matrix is associated with the affine quantum algebra U q ( sl(2)). The explicit formulas for boundary K-matrices for spins s = 1/2, 1 are well known. We derive difference equations for the generating function of matrix elements of the K-matrix for any spin s and solve them in terms of hypergeometric functions. As a result we derive the explicit formula for matrix elements of the K-matrix for arbitrary spin. In the lower-and upper-triangular cases, the K-matrix simplifies and reduces to simple products of q-Pochhammer symbols.then we obtain K J (y) J J = 1 by using the definition (3.13) of the Φ function. Applying the q-Vandermonde summation formula (A.7) it is easy to check that J j=0 K J (y) l j = 1, (5.5) J j=0 K J (y) l j = J l=0(−vt) l (q −2J ; q 2 ) l (q 2 ; q 2 ) l = (−vtq −2J ; q 2 ) J , (7.19)

Generic triangular solutions of the reflection equation: ${U}_{q}\left(\hat{s{l}_{2}}\right)$ case

“…n−1 ) [21] (see also, [22]) and the rational case Y (sl 2 ) [9] are great cues for this. = Eq cH + a(1 − q −2 )q 2b ∞ j=1 (a(1 − q −2 )F q bH ) j−1 (q −2 ; q −2 ) j (q −2(b+c) ; q −2 ) j Cq (b+c)H − (q −2(b+c−1) ; q −2 ) j q (b+c−1)H−1 + (q −2(b+c+1) ; q −2 ) j q (b+c+1)H+1 (q − q −1 ) 2 , (A15)…”

Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes

“…Finally it would be interesting to study the relation to the stochastic R-matrix approach in [23] and the stochastic K-matrix that has been recently obtained in [62] in connection to Baxter Q-operators as discussed at the end of Section 2.4. and exchanged the sums in the second step.…”

Integrable boundaries for the q-Hahn process

“…(3.8) They obey the unitarity relations L a, (x)L ,a (x −1 ) = I. The R-matrix for the fundamental transfer matrix (3.2) is expressed through its components in the same article [2]. It is written in terms of a generalised hypergeometric function as…”

“…From the latter statement it then follows that the left and right K-matrices are determined by the same equation (3.4). The crossing relations have been given in [2]. The first one reads…”

“…We end this section with the boundary Yang-Baxter equation (3.4) in component form as given in [2]. The equations below can be obtained taking coefficients in spectral parameter x:…”