(Z N×) n−1 generalization of the chiral Potts model

Construction ofR-matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$

“…When J = 1 the expression (6.5) further reduces to the trigonometric n-state R-Matrix [29,30]. We shall also use a twisted version of this Rmatrix [31] which we give using notations of [32] …”

“…Using (7.15) for general J and (6.5) one can write it in terms of Weil generators (6.21) similar to (6.23) It satisfies the algebra 20) where S 1,2 (λ) is given by (7.1) with I = J = 1. This L-operator was first obtained in [32] in a slightly different form. The root of unity condition q N = 1 used there does not affect the local structure of the algebra (7.20).…”

Quantum geometry of three-dimensional lattices

“…3 are related by (17)). The map (22) contains the quantities k, a, a * , defined in (19), which now become operators. For definiteness, assume that the non-commuting factors in (19) are ordered exactly as written.…”

“…(ii) there exists a quantum version of the map (22), which acts as an automorphism of the tensor cube of the q-oscillator algebra (53). The formulae (22) for the quantum map stay exactly the same, but the relation (20) should be replaced by either of the two relations on the second line of (53), for instance, k 2 = q (1 − a * a).…”

“…So far only two different (but related) examples of such correspondence have been constructed. The first example, connected with the generalized chiral Potts model [22,23], was found in [16]. The second example was recently found in [21].…”