On the automorphisms of quantum n-spaces and quantum n-tori

A multiparameter quantum affine space \(\mathcal {O}_{\mathfrak q} \) ${\mathcal{O}}_{\mathfrak{q}}$ of rank n is the \(\mathbb {F}\) $\mathbb{F}$ -algebra generated by the indeterminates \(X_1, \ldots , X_n\) $X_1,\dots ,X_n$ satisfying \(X_iX_j = q_{ij} X_jX_i \ (1 \le i < j \le n)\) $X_i{X}_j=q_{\mathit{ij}}{X}_j{X}_i(1\le i<j\le n)$ where \(q_{ij}\) $q_{\mathit{ij}}$ are nonzero scalars in \(\mathbb {F}^*\) ${\mathbb{F}}^{\ast }$ . Necessary and sufficient conditions on the multiparameters \(q_{ij}\) $q_{\mathit{ij}}$ are obtained so that the the only \(\mathbb {F}\) $\mathbb{F}$ -automorphisms are the trivial ones arising from the action of the torus \((\mathbb {F}^*)^n\) ${({\mathbb{F}}^{\ast })}^n$ . We show that the automorphism group is trivial provided that the subgroup \(\Lambda \) $\mathrm{\Lambda }$ of \(\mathbb {F}^*\) ${\mathbb{F}}^{\ast }$ generated by the \(q_{ij}\) $q_{\mathit{ij}}$ has rank at least \(\left( {\begin{array}{c}n - 1\\ 2\end{array}}\right) + 1\) $\left(\begin{array}{c}n-1\\ 2\end{array}\right)+1$ and construct counterexamples when rank is equal to \(\left( {\begin{array}{c}n - 1\\ 2\end{array}}\right) \) $\left(\begin{array}{c}n-1\\ 2\end{array}\right)$ . For the multiparameter quantum torus \(\widehat{\mathcal {O}}_{\mathfrak q} \) ${\widehat{\mathcal{O}}}_{\mathfrak{q}}$ of rank n an important ingredient of the \(\mathbb {F}\) $\mathbb{F}$ -automorphism group is the so-called non-scalar automorphism group, whose calculation is a known to be a hopeless problem. We give examples of calculation of this group for \(n = 4\) $n=4$ and give a general result in this direction. An important case of a quantum torus is when it is hereditary, that is, has global dimension one. This is obtained, for example, when the subgroup \(\Lambda \) $\mathrm{\Lambda }$ has the maximal possible rank \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) $\left(\begin{array}{c}n\\ 2\end{array}\right)$ . We construct examples of hereditary quantum tori with small \(\Lambda \) $\mathrm{\Lambda }$ -rank.