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Free symmetric quantum group

By a free symmetric quantum group one means any element of the one-parameter sequence $(S_N^+)_{N\in \N}$ of compact (matrix) quantum groups defined by Wang in [Wang98] under the name quantum permutation groups. Each $S_N^+$ is a free counterpart of the symmetric group $S_N$ of the corresponding dimension.

Definition

Given $N\in \N$ , the free symmetric quantum group $S_N^+$ (or quantum permuation group on $N$ symbols) is the compact matrix quantum group $(C(S_N^+),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) universal C*-algebra

$C(S_N^+)\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,\forall_{i,j=1}^N:u_{i,j}^2=u_{i,j}=u_{i,j}^\ast, \, {\textstyle\sum_{k=1}^N} u_{i,k}={\textstyle\sum_{l=1}^N} u_{l,j}=1\big\rangle,$