THERMODYNAMICS OF THE U(Q)(XR(1)) BETHE ANSATZ SYSTEM WITH Q A ROOT OF UNITY

The Bethe ansatz equations connected to the U(q)(X(r)(1)) algebra are studied at q = exp(2 pii/(l + g)), with l being a level and g the dual Coxeter number. Based on a ''string hypothesis'' in the thermodynamic limit, the central charges relevant to the X(r)(1) restricted solid-on-solid models and their fusion hierarchies are determined in two critical regimes. The calculation leads naturally to a generalized conjecture on the Rogers dilogarithmic function identity for an arbitrary pair (X(r)(1), l), which includes the earlier one for the simply-laced cases X(r)(1) = A(r)(1), D(r)(1) and E6,7,8(1).