====== 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 group|compact matrix quantum groups]] introduced by Banica and Speicher in [(:ref:BanSp09)]. Each $O_N^\ast$ interpolates the [[wp>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) [[wp>Universal_C*-algebra|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 [[wp>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_quantum_group|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 [[category_of_pair_partitions_with_even_distances_between_legs|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 =====

[( :ref:BanSp09 >>
author:  Banica, Teodor and Speicher, Roland
title:   Liberation of orthogonal Lie groups
year:    2009
journal: Advances in Mathematics
volume:  222
issue:   4
pages:   1461--150
url:     https://doi.org/10.1016/j.aim.2009.06.009
archivePrefix: arXiv
eprint   :0808.2628
)]