On the classification of rational quantum tori and the structure of their automorphism group

TL;DR

This paper classifies rational quantum tori by providing a normal form and analyzes their automorphism groups, showing the sequence splits for two-dimensional cases over any field, advancing understanding of their algebraic structure.

Contribution

It introduces a normal form for rational n-dimensional quantum tori and proves the splitting of the automorphism group sequence for the 2D case over any field.

Findings

Normal form for rational quantum tori established

Automorphism group sequence splits in 2D case

Results hold over any field

Abstract

An n-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.