The functional relations of the transfer matrices of fusion hierachies for sixand eight-vertex models with open boundary conditions have been presented in this paper. We have shown the su(2) fusion rule for the models with more general reflection boundary conditions, which are represented by off-diagonal reflection matrices. Also we have discussed some physics properties which are determined by the functional relations. Finally the intertwining relation between the reflection K matrices for the vertex and SOS models is discussed.to appear in Nucl. Phys. B.Great progress has been made on the study of integrable models in statistical mechanics and quantum field theory. It is very clear that the important methematical structure that ensures the exactly solvablity of these models is governed by the YangBaxter equations if the models sit on square lattices with periodic boundary conditions. Recently there has been interest in studying integrable systems with open boundary conditions. The open boundary conditions are described by boundary reflection matrices satisfying reflection equations [1,2] (boundary Yang-Baxter equations), which ensures the exactly solvablity of the models with the open boundary conditions together with the Yang-Baxter equations. The boundary reflection matrices have been found for many integrable systems, in particular for the six-vertex and eight-vertex models in [3,4,5] (also see [6,7] for related works). The eigenvalues of the transfer matrices have been solved for the six-vertex model or A(1) 1 invariant chain [2,8,13] (also see [16] for related work), Izergin-Korepin vertex model or A (2) 2 invariant chain [9,10], U q (spl(2, 1))-invariant t-J model [11,12], A(1) n invariant chain [6] and A (2) 2n invariant chain [14]. These exact solutions are constructed by the Bethe ansatz only for diagonal reflection matrices. The integrable 1 Email: zhouy@maths.anu.edu.au 2 On leave of absence from Institute of Modern Physics, Northwest University, Xian 710069, China 1 systems with non-diagonal reflection matrices are more difficult to solve and not many results have been obtained for such systems.The fusion procedure has been shown very useful in studying two-dimensional integrable models. Similar to the method to fuse the R matrix of the Yang-Baxter equation [18], the fusion procedure for the reflection matrix K of the reflection equation has been presented in [15]. The fused R-and K-matrices generate some new integrable models with the open boundaries based on the elementary model. The corresponding fused transfer matrices of fusion hierarchies are related through the functional relations, which can be shown by the fusion. In [24] the functional relations of transfer matrices for the six-vertex model with diagonal reflection matrices have been shown and thus the finite size corrections to the transfer matrices have been obtained by solving the functional relations. The functional relations for the six-vertex model and eight-vertex model with non-diagonal reflection matrices have not been fo...

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We present new diagonal solutions of the reflection equation for elliptic solutions of the star-triangle relation. The models considered are related to the affine Lie algebras An . We recover all known diagonal solutions associated with these algebras and find how these solutions are related in the elliptic regime. Furthermore, new solutions of the reflection equation follow for the associated vertex models in the trigonometric limit.

The stereodirecting effect of C5-ester functions on the glycosylation stereoselectivity of 3-deoxy-d- manno-oct-2-ulosonic acid (Kdo) ethyl thioglycoside donors is presented. The coupling of 5- O-arylcarbonyl or acetyl protected Kdo thioglycosides with acceptors proceeds in an α-selective and high-yielding manner, leading to formation of α-linked Kdo glycosides products. On the other hand, the glycosylation stereoselectivity of the 5- O-2-quinolinecarbonyl (Quin) or 4-nitropicoloyl substituted Kdo thioglycoside donors is switchable: (1) The glycosylation of the 5- O-Quin carrying Kdo donors with primary glycosyl acceptors shows complete β-stereoselectivity, furnishing the corresponding β-glycosides in good-to-excellent yield. (2) The stereochemical outcome of the secondary acceptors with these Kdo donors is determined mainly by the stereoelectronic nature of the acceptor. Only or predominant α anomeric products are obtained when the Kdo donors couple with the disarmed or highly crowded secondary carbohydrate acceptors, while the selectivity may switch to predominant β in the glycosylation of the 5- O-4-nitropicoloyl carrying donor with more reactive secondary alcohols. The synthetic use of the newly developed Kdo donors 1c and 7b has been demonstrated by facile preparation of a structurally unique trisaccharide motif 19 which possesses both α- and β-Kdo glycosidic bonds.

The fused six-vertex models with open boundary conditions are studied. The Bethe ansatz solution given by Sklyanin has been generalized to the transfer matrices of the fused models. We have shown that the eigenvalues of transfer matrices satisfy a group of functional relations, which are the su(2) fusion rule held by the transfer matrices of the fused models. The fused transfer matrices form a commuting family and also commute with the quantum group U q [sl (2)]. In the case of the parameter q h = −1 (h = 4, 5, · · ·) the functional relations in the limit of spectral parameter u → i∞ are truncated. This shows that the su(2) fusion rule with finite level appears for the six vertex model with the open boundary conditions. We have solved the functional relations to obtain the finite-size corrections of the fused transfer matrices for low-lying excitations. From the corrections the central charges and conformal weights of underlying conformal field theory are extracted. To see different boundary conditions we also have studied the six-vertex model with a twisted boundary condition.hep-th/9502053; to appear in NPB.1 Email: ykzhou@mundoe.maths.mu.oz.au before 30/04/95.