quantum group* $O_n^{\varepsilon}$ via the following universal $C^*$-algebra $$C(O_n^{\varepsilon})=C^*(u_{ij}, i,j=1,\ldots,n\;|\; u_{ij}=u_{ij}^*, u \textnormal{ is orthogonal}, R^{{\varepsilon}}),$$ where the relations $R^{\varepsilon}$ are defined by: $$u_{ik}u_{jl}=\begin{cases}
u_{jl}u_{ik}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=1$}\\
u_{jk}u_{il}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=0$}\\
u_{il}u_{jk}&\text{if  ${\varepsilon}_{ij}=0$ and ${\varepsilon}_{kl}=1$}
\end{cases}$$

We refer to Proposition \[PropRefinedOn\] for further relations which are implied by the above ones.

\[LemOnQG\] The ${\varepsilon}$-orthogonal quantum group $O_n^{\varepsilon}$ is a quantum group indeed, i.e. the $C^*$-algebra $C(O_n^{\varepsilon})$ gives rise to a compact matrix quantum group in Woronowicz’s sense.

According to Woronowicz’s axioms, all we have to prove is that the map $\Delta:C(O_n^{\varepsilon})\to C(O_n^{\varepsilon})\otimes C(O_n^{\varepsilon})$ with $u_{ij}\mapsto u_{ij}':=\sum_k u_{ik}\otimes u_{kj}$ is a $*$-homomorphism, i.e. that the elements $u_{ij}'\in C(O_n^{\varepsilon})\otimes C(O_n^{\varepsilon})$ satisfy the relations of the $u_{ij}\in C(O_n^{\varepsilon})$. Self-adjointness and orthogonality of $u'$ is easy to see, so it remains to show that the relations $R^{\varepsilon}$ are fulfilled for the $u_{ij}'$. Consider first ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=1$. Then we have: $$\begin{aligned}
u_{ik}'u_{jl}'
&=\sum_{p,r:\, {\varepsilon}_{pr}=1} u_{ir}u_{jp}\otimes u_{rk}u_{pl}+\sum_{p,r:\,
{\varepsilon}_{pr}=0} u_{ir}u_{jp}\otimes u_{rk}u_{pl}\\
&=\sum_{p,r:\, {\varepsilon}_{pr}=1} u_{jp}u_{ir}\otimes u_{pl} u_{rk}+\sum_{p,r:\,
{\varepsilon}_{pr}=0} u_{jr}u_{ip}\otimes u_{rl}u_{pk}\\
&=\sum_{p,r} u_{jp}u_{ir}\otimes u_{pl}u_{rk}\\
&=u_{jl}'u_{ik}'\end{aligned}$$ Consider now ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=0$. Then we have: $$\begin{aligned}
u_{ik}'u_{jl}'
&=\sum_{p,r:\, {\varepsilon}_{pr}=1} u_{ir}u_{jp}\otimes u_{rk}u_{pl}+\sum_{p,r:\,
{\varepsilon}_{pr}=0} u_{ir}u_{jp}\otimes u_{rk}u_{pl}\\
&=\sum_{p,r:\, {\varepsilon}_{pr}=1} u_{jp}u_{ir}\otimes u_{pk} u_{rl}+\sum_{p,r:\,
{\varepsilon}_{pr}=0} u_{jr}u_{ip}\otimes u_{rk}u_{pl}\\
&=\sum_{p,r} u_{jp}u_{ir}\otimes u_{pk}u_{rl}\\
&=u_{jk}'u_{il}'\end{aligned}$$ The case ${\varepsilon}_{ij}=0$ 