RANK OF QUANTIZED UNIVERSAL ENVELOPING-ALGEBRAS AND MODULAR-FUNCTIONS

We compute an intrinsic rank invariant for quasitriangular Hopf algebras in the case of general quantum groups U(q)(g). As a function of q the rank has remarkable number theoretic properties connected with modular covariance and Galois theory. A number of examples are treated in detail, including rank (U(q)(su(3))) and rank (U(q)(e8)). We briefly indicate a physical interpretation as relating Chern-Simons theory with the theory of a quantum particle confined to an alcove of g.