Automorphisms of Quantum Polynomials

Abstract

An important step in the determination of the automorphism group of the quantum torus of rank $n$ (or twisted group algebra of ${\mathbb{Z}}^n$) is the determination of its so-called non-scalar automorphisms. We present a new algorithimic approach towards this problem based on the bivector representation ${\wedge }^2:\mathrm{GL}(n,\mathbb{Z})\to \mathrm{GL}((\genfrac{}{}{0ex}{}{n}{2}),\mathbb{Z})$ of $\mathrm{GL}(n,\mathbb{Z})$ and thus compute the non-scalar automorphism group $\mathrm{Aut}({\mathbb{Z}}^n,\lambda )$ in several new cases. As an application of our ideas we show that the quantum polynomial algebra (multiparameter quantum affine space of rank $n$) has only scalar (or toric) automorphisms provided that the torsion-free rank of the subgroup generated by the defining multiparameters is no less than $\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)+1$ thus improving an earlier result. We also investigate the question: when is a multiparameter quantum affine space free of so-called linear automorphisms other than those arising from the action of the $n$-torus ${({\mathbb{F}}^{\ast })}^n$.