Some Noncommutative Geometric Aspects of Su Q (2)

We study the 3D-calculus on SUq(2) (c.f. [W1]) from the point of view of noncommutative Riemannian geometry as formulated in [F1]. In particular, we show how we can obtain the Haar state from the ”Laplacian” on SUq(2) using a formula very similar to what is used in [F1] but with an appropriate modification. Furthermore, we recast the 3D-calculus along the line of [F1], showing that the complex of forms as defined by Woronowicz in [W1] is isomorphic with the complex obtained from the standard construction of noncommutative Riemannian geometry as in [F1]. Our calculations lead us to conjecture how a generalization of the existing formulation of noncommutative Riemannian geometry may be done in order to accommodate a large class