have been studied extensively under names such as “right angled Artin groups”, “free partially commutative groups”, “trace groups”, “graph groups”, “Cartier-Foata monoids”, “trace monoids” etc, see for instance [@Cha; @Foata] or the references in [@SpW]. It is also linked to the following Coxeter groups (see Def \[DefCox\]): $$\langle a_1,\ldots,a_n\;|\; (a_ia_j)^{m_{ij}}=e\rangle,\qquad 
m_{ij}=\begin{cases} 1 &\textnormal{ if } i=j\\2 &\textnormal{ if } {\varepsilon}_{ij}=1\\\infty &\textnormal{ if }{\varepsilon}_{ij}=0\end{cases}$$

Main results
============

Our main result about the ${\varepsilon}$-sphere $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$ (or rather about its associated $C^*$-algebra $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$) is the following.

The ${\varepsilon}$-spheres are noncommutative as soon as ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$ and we have $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}\neq S_{{ {\mathbb R} },{\varepsilon}'}^{n-1}$ for ${\varepsilon}\neq{\varepsilon}'$. In particular, $$S_{{ {\mathbb R} }}^{n-1}\subsetneq S_{{ {\mathbb R} },{\varepsilon}}^{n-1}\subsetneq S_{{ {\mathbb R} },+}^{n-1}$$ for ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$ and ${\varepsilon}\neq{\varepsilon}_{\operatorname{free}}$.

Here, $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}\neq S_{{ {\mathbb R} },{\varepsilon}'}^{n-1}$ means that there is no isomorphism from $C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1})$ to $C(S_{{ {\mathbb R} },{\varepsilon}'}^{n-1})$ mapping $x_i\mapsto x_i$. The results on $O_n^{\varepsilon}$ may be summarized as follows.

The quantum groups $O_n^{\varepsilon}$ are no groups as soon as ${\varepsilon}\neq{\varepsilon}_{\operatorname{comm}}$ and we have $O_n^{\varepsilon}\neq O_n^{{\varepsilon}'}$ for ${\varepsilon}\neq{\varepsilon}'$. The quantum group $O_n^{\varepsilon}$ acts maximally on the ${\varepsilon}$-sphere.

Note that while the ${\varepsilon}$-sphere $S_{{ {\mathbb R} 