Group-theoretical hyperoctahedral easy orthogonal quantum groups

The group-theoretical hyperoctahedral easy orthogonal quantum groups are a two-parameter family of compact matrix quantum groups introduced by Raum and Weber and classified combinatorially in [RaWe14] and algebraically in [RaWe15]. The first parameter is the matrix dimension $N\in \N$ of the fundamental corepresentation. The second one runs through all strongly symmetric reflection groups on $N$ generators excepting $\Z_2$ and the trivial group. They are particular examples of the more general class of group-theoretical quantum groups.

Definition

For every $N\in \N$ let $\Z_2^{\ast N}$ be the free product group of $N$ many copies of thy cyclic group $\Z_2\equiv \Z/2\Z$ of order $2$ and for every $k\in\{1,\ldots,N\}$ let $a_k$ be the image of $1\equiv 1+2\Z$ under the embedding of $\Z_2$ as the $k$ -th free factor in $\Z_2^{\ast N}$ . Then, $\{a_1,\ldots,a_N\}$ generates $\Z_2^{\ast N}$ and any group endomorphism of $\Z_2^{\ast N}$ is uniquely determined by its restriction to $\{a_1,\ldots,a_N\}$ . The strong symmetric semigroup $\mathrm{sS}_N$ is the subsemigroup of the semigroup $\mathrm{End}(\Z_2^{\ast N})$ of group endomorphisms of $\Z_2^{\ast N}$ generated by the endomorphisms defined by $a_k\mapsto a_{i(k)}$ for all $k\in \{1,\ldots,N\}$ for all mappings $i:\{1,\ldots,N\}\to\{1,\ldots,N\}$ . In other words, $\mathrm{sS}_N$ is given by all identifications of letters in words in an alphabet of $N$ letters from $\Z_2$ .

A strongly symmetric reflection group (on $N$ generators) is now a quotient group of $\Z_2^{\ast N}$ by a normal subgroup $A$ which is invariant under the action of $\mathrm{sS}_N$ , the latter condition meaning $\varphi(w)\in A$ for all $w\in A$ and $\varphi\in \mathrm{sS}_N$ .

Given $N\in \N$ and a strongly symmetric reflection group $\Z^{\ast N}/A$ on $N$ generators other than $\Z_2$ and the trivial group, the group-theoretical hyperoctahedral easy orthogonal quantum group with parameter $\Z^{\ast N}/A$ (for dimension $N$ ) is the compact matrix quantum group $(C(H_N^{<A>}),u)$ 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

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

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

${\color{white}C(H_N^{<A>})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,k,l=1}^N: u_{i,j}^2u_{k,l}=u_{k,l}u_{i,j}^2,$

${\color{white}C(H_N^{<A>})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{m\in \N}:\, \forall_{t_1,\ldots,t_m=1}^N: \, a_{t_1}\ldots a_{t_m}\in A \Rightarrow$

${\color{white}C(H_N^{<A>})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} \forall_{i_1,j_1,\ldots,i_m,j_m=1}^N: (\forall_{p,q=1}^m: i_p=i_q\Leftrightarrow j_p=j_q\Leftrightarrow t_p=t_q)\Rightarrow$

${\color{white}C(H_N^{<A>})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,} u_{i_1,j_1}u_{i_2,j_2}\ldots u_{i_m,j_m}=u_{i_1,j_1}^2u_{i_2,j_2}^2\ldots u_{i_m,j_m}^2\big\rangle,$

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 $H_N^{<A>}$ imply in particular for all $i,j=1,\ldots,N$ that $u_{i,j}^2$ is a projection and that $\sum_{l=1}^N u_{i,l}^2=\sum_{l=1}^N u_{l,j}^2=1$ .

Another characterization of $H_N^{<A>}$ is given by writing it as a semi-direct product with its diagonal subgroup [RaWe15], which is precisely the strongly symmetric reflection group $\Z^{\ast N}/A$ :

$C(H_N^{<A>})\cong C^\ast(\Z^{\ast N}/A)\bowtie C(S_N),$

where $C^\ast(\Z^{\ast N}/A)$ is the group C*-algebra of $\Z^{\ast N}/A$ and where $C(S_N)$ denotes the continuous functions over the symmetric group of dimension $N$ (considered as the subgroup of $\mathrm{GL}(\C,N)$ given by all permutation matrices).

Allowing also $\Z_2$ and the trivial group in the above definition yields the two group-theoretical non-hyperoctahedral easy orthogonal quantum groups $S_N'$ and $S_N$ , the modified symmetric group and the symmetric group $S_N$ .

Basic Properties

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

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

Representation theory

Cohomology

Related quantum groups

References

[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.

[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.

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