Commutative algebraic monoid structures on affine spaces

Abstract

We study commutative associative polynomial operations ${\mathbb{A}}^n\times {\mathbb{A}}^n\to {\mathbb{A}}^n$ with unit on the affine space ${\mathbb{A}}^n$ over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

19 pages