Free modified symmetric quantum group

By a free modified symmetric quantum group one means any element of the one-parameter sequence $(S_N^{\prime +})_{N\in \N}$ of compact matrix quantum groups defined by Banica and Speicher in [BanSp09]. Each $S_N^{\prime+}$ is a free counterpart of the modified symmetric group $S_N'$ of the corresponding dimension $N$ .

Definition

Given $N\in \N$ , the free modified symmetric quantum group $S_N^{\prime +}$ is the compact matrix quantum group $(C(S_N^{\prime+}),u)$ 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

$C(S_N^{\prime +})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u,\, uu^t=u^tu=I_N\otimes 1,\,\forall_{i,j,k=1}^N:i\neq j\Rightarrow u_{i,k}u_{j,k}=u_{k,i}u_{k,j}=0, \, {\textstyle\sum_{l=1}^N} u_{i,l}={\textstyle\sum_{l=1}^N} u_{l,j}\big\rangle,$

where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times\!N$ -matrix and where $1$ is the unit of the universal $C^\ast$ -algebra.

The definition of $S_N^{\prime+}$ is often summarized by saying that it is the compact $N\!\times\!N$ -matrix quantum group whose fundamental corepresentation matrix $u$ is magic'.

Basic Properties

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

If $I$ denotes the closed two-sided ideal of $C(S_N^{\prime +})$ 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 $C(S_N^{\prime +})/I$ is isomorphic to the $C^\ast$ -algebra $C(S_N')$ of continuous functions on the modified symmetric group $S_N'$ , the latter interpreted as the subgroup $\{\pm P\,\vert\, P\in S_N\}$ of $\mathrm{GL}(N,\C)$ given by signed permutation matrices. Hence, $S_N^{\prime +}$ is a compact quantum supergroup of $S_N'$ .

The free modified symmetric quantum groups $(S_N^{\prime +})_{N\in \N}$ 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 of non-crossing partitions of even size that induces the corepresentation categories of $(S_N^{\prime +})_{N\in \N}$ . Its canonical generating set is $\{\fourpart,\singleton\otimes \singleton\}$ .

Representation theory

Cohomology

Related quantum groups

References

[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.

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