Rank of quantized universal enveloping algebras and modular functions (Q802725)

The rank of a finite-dimensional quasitriangular Hopf algebra is a natural generalization of the order of a finite group. In the paper the natural extension of the rank to infinite-dimensional quasitriangular Hopf algebras \(U_ q({\mathfrak g})\)- ``quantum groups'' - is completed for \({\mathfrak g}=su(3)\), su(4), \(e_ 8\). Conjectorially it is the ratio of the product of some \(\theta\)-functions over \(\prod_{\alpha >0}(1-q^{- 2(\alpha,\rho)})\), where \(\alpha\) is a root, \(\rho\) the halfsum of positive roots, at least, for generic q. Numerous consequences of the modular invariance are indicated.