Hyperoctahedral series

The hyperoctahedral series is a family $(H_N^{(s)})_{N,s\in \N,\,s\geq 3}$ of compact matrix quantum groups introduced by Banica, Curran and Speicher in [BanCuSp10]. Each $H_N^{(s)}$ interpolates the hyperoctahedral group $H_N$ and the half-liberated hyperoctahedral quantum group $H_N^{\ast}$ of the corresponding dimension $N$ .

Definition

Given $N\in \N$ and $s\in\N$ with $s\geq 3$ , the quantum group $H_N^{(s)}$ of the hyperoctahedral series with parameter $s$ for dimension $N$ is the compact matrix quantum group $(C(H_N^{(s)}),u)$ 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

$C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u, \, uu^t=u^tu=I_N\otimes 1\,$

${\color{white}C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,k=1}^N: i\neq j\Rightarrow u_{i,k}u_{j,k}=u_{k,i}u_{k,j}=0,$

${\color{white}C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall a,b,c\in \{u_{i,j}\}_{i,j=1}^n: acb=bca,$

${\color{white}C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall a,b\in \{u_{i,j}\}_{i,j=1}^n: (s\text{ odd} \Rightarrow (ab)^{\frac{s-1}{2}}a=(ba)^{\frac{s-1}{2}}b),\, (s\text{ even}\Rightarrow (ab)^{\frac{s}{2}}=(ba)^{\frac{s}{2}})\big\rangle,$

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 can also be expressed by saying that the fundamental corpresentation matrix $u$ of $H_N^{(s)}$ is cubic and satisfies the half-commutation relations and $s$ -mixing relations.

Had one allowed $s=2$ in the definition, one would have obtained the hyperoctahedral group $H_N^{(2)}\colon\hspace{-0.66em}=H_N$ . However, this quantum group is not half-liberated in contrast to $H_N^{(s)}$ for all $s\geq 3$ .

Sometimes, the half-liberated hyperoctahedral quantum group $H_N^{\ast}$ is considered an element of the hyperoctahedral series via the definition $H_N^{(\infty)}\colon\hspace{-0.66em}= H_N^{\ast}$ .

The quantum groups of the hyperoctahedral series are group-theoretical hyperoctahedral orthogonal easy quantum groups and can therefore be written as semi-direct products with their diagonal subgroups [RaWe15]:

$C(H_N^{(s)})\cong C^\ast\langle \{a_i\}_{i=1}^n \,\vert\, \forall_{i,j=1}^n: a_i^2=1,\, (a_ia_j)^s=1,\,\forall_{i,j,k=1}^n: a_i a_k a_j=a_j a_k a_i \rangle\bowtie C(S_N)$

for all $s,N\in \N$ with $s\geq 3$ , where $C(S_N)$ denotes the continuous function over the symmetric group of dimension $N$ (considered as the subgroup of $\mathrm{GL}(\C,N)$ given by all permutation matrices).

Basic Properties

The fundamental corepresentation matrix $u$ of $H_N^{(s)}$ is in particular orthogonal. Hence, $H_N^{(s)}$ is a compact quantum subgroup of the free orthogonal quantum group $O_N^+$ .

Moreover, $u$ is also cubic especially, implying that $H_N^{(s)}$ is a compact quantum subgroup of the free hyperoctahedral quantum group $H_N^{+}$ , the free counterpart of the hyperoctahedral group $H_N$ .

Per definition the fundamental corepresentation matrix $u$ of $H_N^{(s)}$ is not only cubic but also satisfies the half-commutation relations. For that reason, $H_N^{(s)}$ is a compact quantum subgroup of the half-liberated hyperoctahedral quantum group $H_N^{\ast}$ .

If $I$ denotes the closed two-sided ideal of $C(H_N^{(s)})$ 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 $C(H_N^{(s)})/I$ is isomorphic to the $C^\ast$ -algebra $C(H_N)$ of continuous functions on the hyperoctahedral group $H_N$ , the subgroup of $\mathrm{GL}(N,\C)$ given by orthogonal matrices with integer entries. Hence, $H_N^{(s)}$ is a compact quantum supergroup of $H_N$ .

For every $s\in \N$ with $s\geq 3$ the quantum groups $(H_N^{(s)})_{N\in \N}$ of the hyperoctahedral series with paraemter $s$ 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 a group-theoretical hyperoctahedral category of partitions that induces the corepresentation categories of $(H_N^{(s)})_{N\in \N}$ . Canonically, it is generated by the set $\{\Pabcabc,\fourpart,h_s\}$ of partitions [RaWe14], where $h_s$ is the partition whose word representation is given by $(ab)^s$ . See also categories of the hyperoctahedral series.

Representation theory

Cohomology

Related quantum groups

References

[BanCuSp10] Banica, Teodor and Curran, Stephen and Speicher, Roland, 2010. Classification results for easy quantum groups. Pacific Journal of Mathematics, 247, pp.1-26.

[RaWe15] Raum, Sven and Weber, Moritz, 2015. Easy quantum groups and quantum subgroups of a semi-direct product quantum group. Journal of Noncommutative Geometry, 9, pp.1261–1293.

[RaWe14] Raum, Sven and Weber, Moritz, 2014. The combinatorics of an algebraic class of easy quantum groups. Infinite Dimensional Analysis, Quantum Probability and related topics, 17.

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