On some algebraic and geometric aspects of the quantum unitary group

TL;DR

This paper investigates algebraic and geometric properties of the quantum group U_q(2), demonstrating non-isomorphism of certain C*-algebras for different q values and constructing a spectral triple with specific properties.

Contribution

It proves non-isomorphism of C(U_q(2)) for non-real and real q, and introduces a torus action along with a spectral triple that is even and 3^+-summable.

Findings

C(U_q(2)) and C(U_{q'}(2)) are non-isomorphic when q is non-real and q' is real

Constructed a torus-equivariant spectral triple for U_q(2)

The Dirac operator is K-homologically nontrivial

Abstract

Consider the compact quantum group $U_q(2)$, where $q$ is a non-zero complex deformation parameter such that $|q|\neq 1$. Let $C(U_q(2))$ denote the underlying $C^*$-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 $C^*$-algebras $C(U_q(2))$ and $C(U_{q^\prime}(2))$ are non-isomorphic. This is in sharp contrast with the case of braided $SU_q(2)$, 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 $(C(U_q(2)),\mathbb{T}^3,\alpha)$. 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.