Free hyperoctahedral quantum group

By a free hyperoctahedral quantum group one means any element of the one-parameter sequence $(H_N^+)_{N\in \N}$ of compact matrix quantum groups defined by Banica, Bichon and Collins in [BanBichColl07]. Each $H_N^+$ is a free counterpart of the hyperoctahedral group $H_N$ of the corresponding dimension $N$ .

Definition

Given $N\in \N$ , the free hyperoctahedral quantum group $H_N^+$ is the compact matrix quantum group $(C(H_N^+),u)$ 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

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

where $\overline u=(u_{i,j}^\ast)_{i,j=1}^N$ is the complex conjugate and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose of $u$ , 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 $H_N^+$ is often equivalently expressed by saying that the fundamental corepresentation matrix $u$ is cubic.

Basic Properties

If $I$ denotes the closed two-sided ideal of $C(H_N^+)$ 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 $C(H_N^+)/I$ is isomorphic to the $C^\ast$ -algebra $C(H_N)$ of continuous functions on the hyperoctahedral group, the latter realized as the group of orthogonal $N\!\times\!N$ -matrices with integer entries. Hence, $H_N^+$ is a compact quantum supergroup of $H_N$ .

The free hyperoctahedral quantum groups $(H_N^+)_{N\in \N}$ 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 of non-crossing partitions with blocks of even size that induces the corepresentation categories of $(H_N^+)_{N\in \N}$ . Its canonical generating partition is $\fourpart$ .

Representation theory

Cohomology

Related quantum groups

References

[BanBichColl07] Banica Teodor and Bichon Julien and Collins Benoit, 2007. The hyperoctahedral quantum group. Journal of the Ramanujan Mathematical Society, 22(4), pp.345–384.

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