On some algebraic and geometric aspects of the quantum unitary group

Consider the compact quantum group \(U_q(2)\) $U_q(2)$ , where q is a non-zero complex deformation parameter such that \(|q|\ne 1\) $|q|\ne 1$ . Let \(C(U_q(2))\) $C(U_q(2))$ denote the underlying \(C^*\) $C^{\ast }$ -algebra of the compact quantum group \(U_q(2)\) $U_q(2)$ . We prove that when q is a non-real complex number and \(q^\prime \) $q^{\prime }$ is real, the underlying \(C^*\) $C^{\ast }$ -algebras \(C(U_q(2))\) $C(U_q(2))$ and \(C(U_{q^\prime }(2))\) $C(U_{q^{\prime }}(2))$ are non-isomorphic. This is in sharp contrast with the case of braided \(SU_q(2)\) $S{U}_q(2)$ , introduced earlier by Woronowicz et al., where q is a non-zero complex deformation parameter. In another direction, on a geometric aspect of \(U_q(2)\) $U_q(2)$ , we introduce torus action on the \(C^*\) $C^{\ast }$ -algebra \(C(U_q(2))\) $C(U_q(2))$ and obtain a \(C^*\) $C^{\ast }$ -dynamical system \((C(U_q(2)),\mathbb {T}^3,\alpha )\) $(C(U_q(2)),{\mathbb{T}}^3,\alpha )$ . Finally, we construct a \(\mathbb {T}^3\) ${\mathbb{T}}^3$ -equivariant spectral triple for \(U_q(2)\) $U_q(2)$ that is even and \(3^+\) $3^{+}$ -summable. It is shown that the Dirac operator is K-homologically nontrivial.