On some algebraic and geometric aspects of the quantum unitary group

Consider the compact quantum group Uq (2), where q is a non-zero complex deformation parameter such that |q| 1. Let C (Uq (2) ) denote the underlying C^*-algebra of the compact quantum group Uq (2). We prove that if q is a non-real complex number and q^ is real, then the underlying C^*-algebras C (Uq (2) ) and C (Uₐ^ (2) ) are non-isomorphic. This is in sharp contrast with the case of braided SUq (2), introduced earlier by Woronowicz et al. , where q is a non-zero complex deformation parameter. In another direction, on a geometric aspect of Uq (2), we introduce torus action on the C^*-algebra C (Uq (2) ) and obtain a C^*-dynamical system (C (Uq (2) ), T³, ). We construct a T³-equivariant spectral triple for Uq (2) that is even and 3^+-summable. It is shown that the Dirac operator is K-homologically nontrivial.