Quantum unitary group $U_{q,\hat{I}˜}(3)$ for complex deformation parameters

Abstract

In this article, we consider a particular Hayashi $R$-matrix that satisfies the Yang-Baxter equation $R_{12}{R}_{13}{R}_{23}=R_{23}{R}_{13}{R}_{12}$. Using the FRT-bialgebra technique and Woronowicz's method of construction, we construct a concrete compact quantum group $U_{q,\hat{I}˜}(3)$ for non zero real $q$ and modulus one complex deformation parameters $\hat{I}{¸}_{ij}$. We then study in detail the irreducible $\ast$-representations of the underlying $C^{\ast }$-algebra $C(U_{q,\hat{I}˜}(3))$, using the representations of the three dimensional noncommutative torus. Also, a monomial basis for the dense Hopf *-algebra $\mathbb{C}[U_{q,\hat{I}˜}(3)]$ is obtained.