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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

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

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 $u$ 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 is to say $acb=bca$ for all $a,b,c\in \{u_{i,j}\}_{i,j=1}^N$ .

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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