Classification of Connected Commutative Locally Nash Groups

TL;DR

This paper classifies connected commutative locally Nash groups, extending previous classifications of abelian Nash manifolds and analyzing their geometric properties such as affineness.

Contribution

It generalizes the classification of low-dimensional Nash groups and determines their affineness and toroidal properties.

Findings

Classification of connected commutative Nash groups achieved

Extension of previous low-dimensional classifications

Identification of affineness and toroidal properties

Abstract

In this article, we classify connected commutative (locally) Nash groups, which is a continuation of our previous work on the classification of abelian Nash manifolds. Our results generalize the classification of the one-dimensional case by Madden-Stanton and the two-dimensional case by Baro-Vicente-Otero. Moreover, we determine the affineness and toroidal affinenese of a connected commutative Nash group.