Some Noncommutative Geometric Aspects of SU_q(2)

We study various noncommutative geometric aspects of the compact quantum group SU_q(2) for positive q (not equal to 1), following the suggestion of Connes and his coauthors [CL, CD] for considering the so-called true Dirac operator. However, it turns out that the method of the above references do not extend to the case of positive (not equal to 1) values of q in the sense that the true Dirac operator does not have bounded commutators with "smooth" algebra elements in this case, in contrast to what happens for complex q of modulus 1. Nevertheless, we show how to obtain the canonical volume form, i.e. the Haar state, using the true Dirac operator.