Algebraic equations determining quantum dimensions

Abstract

We show that a quantum dimension Dq( Lambda ) for a representation rho of Uq(G), a quantized universal enveloping algebra of a compact and simple Lie group G, is computed from the algebraic equations which we found recently in studying 2+1-dimensional Chern-Simons theory. We solve the equations explicitly for the typical examples of all compact and simple Lie groups. This method can be applied to super Lie groups such as SU(m,n) and OSp(m,n).