Automorphisms of Quantum Polynomials

TL;DR

This paper introduces a new algorithmic method to determine non-scalar automorphisms of quantum tori and polynomial algebras, providing new classifications and conditions under which only scalar automorphisms occur.

Contribution

The authors develop an innovative approach using bivector representations to compute automorphism groups and establish conditions for scalar automorphisms in quantum polynomial algebras.

Findings

Computed non-scalar automorphism groups in new cases

Proved quantum polynomial algebras have only scalar automorphisms under certain conditions

Improved previous results on automorphism classifications

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 $\wedge^2 : \mathrm{GL}(n, \mathbb Z) \rightarrow \mathrm{GL}(\binom{n}{2}, \mathbb Z)$ 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 $\binom{n - 1}{2} + 1$ 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 ${(\mathbb F^\ast)}^n$.