On some algebraic and geometric aspects of the quantum unitary group: On some algebraic and geometric aspects: D. Jana.

Consider the compact quantum group U q (2) , where q is a non-zero complex deformation parameter such that | q | ≠ 1 . Let C (U q (2)) denote the underlying C ∗ -algebra of the compact quantum group U q (2) . We prove that when q is a non-real complex number and q ′ is real, the underlying C ∗ -algebras C (U q (2)) and C (U q ′ (2)) are non-isomorphic. This is in sharp contrast with the case of braided S U q (2) , introduced earlier by Woronowicz et al., where q is a non-zero complex deformation parameter. In another direction, on a geometric aspect of U q (2) , we introduce torus action on the C ∗ -algebra C (U q (2)) and obtain a C ∗ -dynamical system (C (U q (2)) , T 3 , α) . Finally, we construct a T 3 -equivariant spectral triple for U q (2) that is even and 3 + -summable. It is shown that the Dirac operator is K-homologically nontrivial. [ABSTRACT FROM AUTHOR]