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 $N$ .

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

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

where $1$ is the unit of the universal $C^\ast$ -algebra.

In other words, the entries $\{u_{i,j}\}_{i,j=1}^N$ of the fundamental corpresentation matrix $u$ of $S_N^+$ are projections, i.e., self-adjoint idempotents, and the entries of each row or column form a partition of unity, i.e., mutually orthogonal projections summing up $1$ (where the orthogonality is to mean $u_{i,j}u_{i,k}=\delta_{j,k}u_{i,j}$ and $u_{i,j}u_{l,j}=\delta_{i,l}u_{i,j}$ for all $i,j,k,l=1,\ldots,N$ as can be shown). Those relations are commonly summarized by saying that $u$ is a magic unitary.

Basic Properties

The fundamental corepresentation matrix $u$ of $S_N^+$ is in particular orthogonal. Hence, $S_N^+$ is a compact quantum subgroup of the free orthogonal quantum group $O_N^+$ .

If $I$ denotes the closed two-sided ideal of $C(S_N^+)$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$ , then $C(S_N^+)/I$ is isomorphic to the $C^\ast$ -algebra $C(S_N)$ of continuous functions on the symmetric group $S_N$ , the latter interpreted as the subgroup of $\mathrm{GL}(N,\C)$ given by all permutation matrices. Hence, $S_N^+$ is a compact quantum supergroup of $S_N$ .

The free symmetric quantum groups $(S_N^+)_{N\in \N}$ are an easy family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a category of partitions. More precisely, it is the category $\mathrm{NC}$ of all non-crossing partitions that induces the corepresentation categories of $(S_N^+)_{N\in \N}$ . The canonical generating set of partitions of $\mathrm{NC}$ is $\{\fourpart,\singleton\}$ .

Representation theory

Cohomology

Related quantum groups

References

[Wang98] Shuzhou Wang, 1998. Quantum Symmetry Groups of Finite Spaces. Communications in Mathematical Physics, 195(1), pp.195–211.

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