Using an eight-state cyclic L-operator related to the Uq( sl (3)) algebra at q2 = -1 we have found an interwining eight-state R-matrix in the "trigonometric" limit. Corresponding commuting family of transfer matrices depends on three independent spectral parameters.
We construct R-matrices (with a multidimensional spectral parameter) that include additive as well as non-additive parameters. They satisfy the colored Yang-Baxter equation. The solutions depend on a set of commuting operators. They change with the representation of the latter. The associated Yang-Baxter algebra and the spectrum of the transfer matrix are also studied.
We construct a Q-matrix for the eight-vertex model at roots of unity for crossing parameter η=2mK/L with odd L, a case for which the existing constructions do not work. The new Q-matrix $\Q$ depends as usual on the spectral parameter and also on a free parameter t. For t=0 $\Q$ has the standard properties. For t=0, however, it does not commute with the operator S and not with itself for different values of the spectral parameter. We show that the six-vertex limit of $\Q(v,t=iK'/2)$ exists.
The Nn(n–1)/2-state R-matrix related to U q( sl (n)) algebra at q2N=1 is presented. Its matrix elements are interpreted as Boltzmann weights of an elementary box of some 2D lattice statistical model and given in terms of [Formula: see text] weights of the "minimal" sl (n) chiral Potts model. The corresponding family of transfer matrices depends on n rapidity variables living on high genus algebraic curves, the latter being defined by n moduli. The Yang-Baxter equation is conjectured to hold.