Commutative algebraic monoid structures on affine spaces

We study commutative associative polynomial operations $\mathbb{A}^n \times \mathbb{A}^n \rightarrow \mathbb{A}^n$ 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. This is a joint work with Sergey Bragin and Yulia Zaitseva.