Quantum Symmetry Groups of Infinite Dimensional Spaces

The $n$-cube graphs (or the $n$-hypercube graphs) $\mathcal{Q}_n$ could be seen as the Cayley graph $\caay(G,\mathcal{S})$ for $G=\bigoplus\limits_r\mathbb Z_{2}^{r^2}$ and $\mathcal{S}:=\{E_{ij}=(e_{op})_{1\le o,p\le n} \ | \ e_{ij}=1 \ \& \ e_{op}=0 \ \text{for} \ o\neq i \ \text{and} \ p\neq j\}$. It is a well-known result that the quantum automorphism group $G_{\qaut}(\mathcal{Q}_n):=H_{n}^{+}$ is isomorphic with the anticommutative orthogonal quantum group $\mathcal{O}(O_{r}^{-1})$, for some $r$ dependent on $n$ (here in this paper we will try to specify $r$) but our knowledge helps, as we talk, the main generators of $G_{\qaut}(\mathcal{Q}_n)$ have not been observed yet. So, in this paper we tried to study them along with the Cayley graphs such as $\caay(\mathbb K_2,\mathcal{S})$ for the free group on two generators $\mathbb K_2$, and the folded $n$-cube graphs $\mathsf{F}\mathcal{Q}_{n}$ and their generalized versions $\prescript{3}{\Delta}{\mathsf{F}\mathcal{Q}_{n}}$. In the last part of this paper, we tried to study the quantum symmetric groups of infinite dimensional spaces, by looking at them as arbitrary matrix spaces with points considered arbitrary matrices!We believe that this way of observation might help providing a partial answer to the open problem on the unit distance graphs and their quantum symmetries!