Commutative algebraic monoid structures on affine spaces

摘要

We study commutative associative polynomial operations A(n) x A(n) -> A(n) with unit on the affine space 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.