Non-group-theoretical hyperoctahedral easy orthogonal quantum groups

The non-group-theoretical hyperoctahedral easy orthogonal quantum groups are a family $(H_N^{[\pi_k]})_{N\in \N,\,k\in \{2,3\ldots,\infty\}}$ of compact matrix quantum groups introduced by Raum and Weber in [RaWe16]. Each $H_N^{[\pi_k]}$ interpolates the free hyperoctahedral quantum group $H_N^+$ and any group-theoretical hyperoctahedral easy orthogonal quantum groups of the corresponding dimension $N$ .

Definition

Given $N\in \N$ and $k\in\N\cup \{\infty\}$ with $k\geq 2$ , the non-group-theoretical hyperoctahedral easy orthogonal quantum group $H_N^{[\pi_k]}$ with parameter $k$ for dimension $N$ is the compact matrix quantum group $(C(H_N^{[\pi_k]}),u)$ 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

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

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

${\color{white}C(H_N^{[\pi_k]})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{m\in \N}: m<k\Rightarrow\forall a,b_1,\ldots,b_{m}\in \{u_{i,j}\}_{i,j=1}^n:$

${\color{white}C(H_N^{[\pi_k]})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} a^2 \ (b_1b_2\ldots b_m)\ (b_m b_{m-1}\ldots b_1)= (b_1b_2\ldots b_m) \ (b_mb_{m-1}\ldots b_1) \ a^2\big\rangle,$

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 defining relations of $H_N^{[ \pi_k]}$ imply in particular for all $i,j=1,\ldots,N$ that $u_{i,j}^2$ is a partial isometry and that $\sum_{l=1}^N u_{i,l}^2=\sum_{l=1}^N u_{l,j}^2=1$ .

The non-group-theoretical hyperoctahedral easy orthogonal quantum groups $(H_N^{[\pi_2]})_{N\in\N}$ with parameter $2$ are also called the square commuting hyperoctahedral quantum groups $(H_N^\diamond)_{N\in \N}$ .

Had one allowed $k=1$ in the definition, one would have obtained the free hyperoctahedral quantum group $H_N^{[\pi_1]}\colon\hspace{-0.66em}=H_N^+$ .

Basic Properties

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

Moreover, $u$ is also cubic especially, implying that $H_N^{[\pi_k]}$ is a compact quantum subgroup of the free hyperoctahedral quantum group $H_N^{+}$ , the free counterpart of the hyperoctahedral group $H_N$ .

For all $k,k'\in \N\cup\{\infty\}$ with $k<k'$ the fundamental corepresentation matrix of $H_N^{[\pi_{k'}]}$ also satisfies the relations of $H_N^{[\pi_k]}$ , meaning that $H_N^{[\pi_k']}$ is a compact quantum subgroup of $H_N^{[\pi_k]}$ . Especially, the sequence $(H_N^{[\pi_{k}]})_{k\in \N\cup\{\infty\}$ forms a descending chain.

Conversely, the fundamental corepresentation matrix of any group-theoretical hyperoctahedral easy orthogonal quantum group satisfies the defining relations of $H_N^{[\pi_k]}$ for any $k\in \N\cup \{\infty\}$ . Hence, $H_N^{[\pi_k]}$ is a compact quantum supergroup of any such quantum group for every $k$ .

For every $k\in \N\cup\{\infty\}$ with $k\geq 2$ the non-group-theoretical hyperoctahedral easy orthogonal quantum groups $(H_N^{[\pi_k]})_{N\in \N}$ with parameter $k$ 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 non-group-theoretical hyperoctahedral category of partitions that induces the corepresentation categories of $(H_N^{[\pi_k]})_{N\in \N}$ . Canonically, it is generated by the partition $\pi_k$ if $k<\infty$ and by the set $\{\pi_{l}\,\vert\,l\in\N\}$ of partitions if $k=\infty$ [RaWe16], where $\pi_l$ is the partition whose word representation is given by $\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_l\mathsf{a}_l\cdots\mathsf{a}_2\mathsf{a}_1\mathsf{a}_1\mathsf{a}_2\cdots\mathsf{a}_l\mathsf{a}_l\cdots\mathsf{a}_2\mathsf{a}_1$ .

Representation theory

Cohomology

Related quantum groups

References

[RaWe16] Raum, Sven and Weber, Moritz, 2016. The full classification of orthogonal easy quantum groups. Communications in Mathematical Physics, 341, pp.751–779.

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