Table of Contents

Square commuting hyperoctahedral quantum group

The square commuting hyperoctahedral quantum groups are a family $(H_N^{\diamond})_{N\in \N}$ of compact matrix quantum groups introduced by Raum and Weber in [RaWe16]. Each $H_N^{\diamond}$ interpolates the free hyperoctahedral quantum group $H_N^+$ and any group-theoretical hyperoctahedral easy orthogonal quantum groups of the corresponding dimension $N$. They are a special instance of non-group-theoretical hyperoctahedral easy orthogonal quantum groups.

Definition

Given $N\in \N$, the square commuting (hyperoctahedral) quantum group $H_N^{\diamond}$ for dimension $N$ is the compact matrix quantum group $(C(H_N^{\diamond}),u)$ 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

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

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

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

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 $H_N^{\diamond}$ 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 square commuting quantum group $H_N^{\diamond}$ is the maximal quantum group of the descending chain $(H_N^{[\pi_k]})_{k\in \{2,\ldots,\infty\}}$ of non-group-theoretical hyperoctahedral easy orthogonal quantum groups of dimension $N$.

Basic Properties

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

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

The fundamental corepresentation matrix of any group-theoretical hyperoctahedral easy orthogonal quantum group satisfies the defining relations of $H_N^{\diamond}$. Hence, $H_N^{\diamond}$ is a compact quantum supergroup of any such quantum group.

The square commuting quantum groups $(H_N^{\diamond})_{N\in \N}$ 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 a non-group-theoretical hyperoctahedral category of partitions that induces the corepresentation categories of $(H_N^{\diamond})_{N\in \N}$. Canonically, it is generated by the partition $\pi_2$ [RaWe16], where $\pi_2$ is the partition whose word representation is given by $\mathsf{a}_1\mathsf{a}_2\mathsf{a}_2\mathsf{a}_1\mathsf{a}_1\mathsf{a}_2\mathsf{a}_2\mathsf{a}_1$.

Representation theory

Cohomology

References

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