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Half-liberated orthogonal quantum group

The half-liberated orthogonal quantum groups are the elements of a sequence $(O_N^\ast)_{N\in \N}$ of compact matrix quantum groups introduced by Banica and Speicher in [BanSp09]. Each $O_N^\ast$ interpolates the orthogonal group $O_N$ and the free orthogonal quantum group $O_N^+$ . It is characterized by the half-commutation relations.

Definition

Given $N\in \N$ , the half-liberated orthogonal quantum group $O_N^\ast$ is the compact matrix quantum group $(C(O_N^\ast),u)$ 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

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

where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate and $u^t=(u_{i,j})_{i,j=1}^N$ is the transpose of $u$ , where $I_N$ is the identity $N\!\times\!N$ -matrix and where $1$ is the unit of the universal $C^\ast$ -algebra.

In words, $O_N^\ast$ is the compact matrix quantum group whose fundamental corepresentation matrix is orthogonal and whose entries satisfy the half-commutation relations (which one calls the relations say $acb=bca$ for all entries $a,b,c$ ). Of course, these latter relations need to be viewed as a generalization of the commutation relations $ab=ba$ satisfied by all entries $a,b$ of the fundamental corepresentation matrix of $O_N$ .

Basic properties

By definition, the fundamental corepresentation matrix $u$ of $O_N^\ast$ is orthogonal. That makes $O_N^\ast$ a compact quantum subgroup of the free orthogonal quantum group $O_N^+$ .

If $I$ denotes the closed two-sided ideal of $C(O_N^\ast)$ generated by the commutation relations $ab=ba$ for all $a,b\in \{u_{i,j}\}_{i,j=1}^N$ , then $C(O_N^\ast)/I$ is isomorphic to the $C^\ast$ -algebra $C(O_N)$ of continuous functions on the orthogonal group $O_N$ . Hence, $O_N^\ast$ is a compact quantum supergroup of $O_N$ .

In conclusion, $O_N\subseteq O_N^\ast\subseteq O_N^+$ for every $N\in \N$ . That explains the name “half-liberated”: The half-liberated orthogonal quantum group $O_N^\ast$ is a “halfway point” between the orthogonal group $O_N$ and its “liberation”, the free orthogonal quantum group $O_N^+$ .

The half-liberated orthogonal quantum groups $(O_N^\ast)_{N\in \N}$ 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 all pair partitions with evenly many crossings, sometimes denoted by $P_o^\ast$ , that induces the corepresentation categories of $(O_N^\ast)_{N\in \N}$ . Its canonical generator partition is $\Pabcabc$ .

Representation theory

Cohomology

Related quantum groups

References

[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.

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