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Half-liberated hyperoctahedral quantum group

The half-liberated hyperoctahedral quantum groups are the elements of a sequence $(H_N^{\ast})_{N\in \N}$ of compact matrix quantum groups introduced by Banica, Curran and Speicher in [BanCuSp10]. Each $H_N^{\ast}$ interpolates the hyperoctahedral group $H_N$ and the free hyperoctahedral quantum group $H_N^{+}$ of the corresponding dimension $N$ .

Definition

Given $N\in \N$ , the half-liberated hyperoctahedral quantum group $H_N^{\ast}$ is the compact matrix quantum group $(C(H_N^{\ast}),u)$ 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

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

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 $H_N^{\ast}$ is cubic and satisfies the half-commutation relations.

Basic Properties

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

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

If $I$ denotes the closed two-sided ideal of $C(H_N^{\ast})$ 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 $C(H_N^{\ast})/I$ 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, $H_N^{\ast}$ is a compact quantum supergroup of $H_N$ .

The half-liberated hyperoctahedral quantum groups $(H_N^{\ast})_{N\in \N}$ 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 partitions with blocks of even size and even distances between legs that induces the corepresentation categories of $(H_N^{\ast})_{N\in \N}$ . Canonically, it is generated by the set $\{\Pabcabc,\fourpart\}$ of partitions.

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.

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