Diagonal subgroup of a compact matrix quantum group

The diagonal subgroup is an algebraic invariant of compact matrix quantum groups introduced by Raum and Weber in [RaWe15].

Definition

Given a compact matrix quantum group $G\cong(C(G),u)$ of dimension $N\in \N$ the diagonal subgroup of $G$ is defined as the unique (classical) group $\Gamma$ such that a group C*-algebra $C^\ast(\Gamma)$ of $\Gamma$ is isomorphic as a $C^\ast$ -algebra to the quotient $C(G)/I$ , where $I$ is the closed two-sided ideal of $C(G)$ generated by the relations $\{u_{i,j}=0\,\vert\, i,j\in\N,\, i\neq j\}$ . It is denoted by $\mathrm{diag}(G)\colon\hspace{-0.66em}=\Gamma$ .