Twisted forms of commutative monoid structures on affine spaces

TL;DR

This paper classifies affine commutative algebraic monoid structures on affine spaces over fields of characteristic zero, providing complete results for dimensions two and three and exploring generalizations to higher dimensions.

Contribution

It offers a full classification of such monoid structures on a0a0^2 and a0a0^3, and extends the analysis to higher dimensions.

Findings

Complete classification on a0a0^2 and a0a0^3.

Descriptions of generalizations to a0a0^n for any n.

Structural insights into affine commutative monoids on affine spaces.

Abstract

In this paper, we study affine commutative algebraic monoid structures on affine spaces over an arbitrary field of characteristic zero. We obtain full classification of such structures on $\mathbb{A}_K^2$ and $\mathbb{A}_K^3$ and describe some generalizations on $\mathbb{A}_K^n$ for any dimension $n$.