Infinitesimal Generators on the Quantum Group SUq(2) in the Classical and Anti-Classical Limit

Abstract

In [5] we found an explicit formula for the infinitesimal generators of white noises on the quantum group SUq(2) in the case q e (-1, 1). In [7] we compared the case q e (-1, 1) with the classical case q = 1 which corresponds to the infinitesimal generators of Levy processes on SU(2). We pointed out that our formula is the perfect analogue of Hunt's formula which describes the classical infinitesimal generators. In the first part of these notes we find Hunt's formula for SUq(2) in the anti-classical case q = -1. This completes our theory of infinitesimal generators on SUq(2) for all q e [-1, 1]. In the second part of these notes we show that SUq(2) is a 'good' quantization of SU(2). We show that not only states on the algebra of functions on SU(2), but even infinitesimal generators may be approximated in a suitable sense by states and infinitesimals generators on SUq(2) for q e (-1, 1). Simultaneously, we show that a similar result is true for SU-1(2).