Uq[sl(n)] Modules and Invariant Integrable n-State Models with Periodic Boundary Conditions

The weights are computed for the Bethe vectors of an RSOS type model with periodic boundary conditions obeying Uq[sl(n)] (q = exp(iπ/r)) invariance. They are shown to be highest weight vectors. The q-dimensions of the corresponding irreducible representations are obtained. In the last years considerable progress has been made on the ”quantum symmetry” of integrable quantum chain models as the XXZ-Heisenberg model and its generalizations. In [1] we constructed an slq(n) invariant RSOS type model with periodic boundary conditions. In the present paper we prove for this model the highest weight property of the Bethe states, calculate the weights and the q-dimensions of the representations and classify the irreducible ones. For the case of open boundary conditions see e.g. [2], [3] and [4]. The model of [1] is defined by the transfer matrix τ = τ (n) where τ (x, x) = trq(T (x, x) = ∑ α q(T )α(x, x ), k = 1, . . . , n. (1) The “doubled” monodromy matrix is given by T (k) 0 (x, x ) = T̃ (k) 0 · T (k) 0 (x, x ) = (R01 . . . R0Nk) · (RNk0(xNk/x ) . . . R10(x1/x )). (2) Supported by DFG, Sonderforschungsbereich 288 ’Differentialgeometrie und Quantenphysik’ e-mail: [email protected] 1 For the purpose of the nested algebraic Bethe ansatz in addition to T (x) = T (x) the monodromy matrices for all k ≤ n are needed. The slq(k) R-matrix is given by R(x) = xR−xPRP, R = ∑