On some algebraic and geometric aspects of the quantum unitary group

Abstract

Consider the compact quantum group $U_q(2)$, where $q$ is a non-zero complex deformation parameter such that $|q|\ne 1$. Let $C(U_q(2))$ denote the underlying $C^{\ast }$-algebra of the compact quantum group $U_q(2)$. We prove that if $q$ is a non-real complex number and $q^{\prime }$ is real, then the underlying $C^{\ast }$-algebras $C(U_q(2))$ and $C(U_{q^{\prime }}(2))$ 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 $C^{\ast }$-algebra $C(U_q(2))$ and obtain a $C^{\ast }$-dynamical system $(C(U_q(2)),{\mathbb{T}}^3,\alpha )$. We construct a ${\mathbb{T}}^3$-equivariant spectral triple for $U_q(2)$ that is even and $3^{+}$-summable. It is shown that the Dirac operator is K-homologically nontrivial.