On polynomial products in nilpotent and solvable Lie groups

Abstract:

We are dealing with Lie groups $G$ which are diffeomorphic to ${\mathbb R}^n$, for some $n$. After identifying $G$ with ${\mathbb R}^n$, the multiplication on $G$ can be seen as a map $\mu :{\mathbb R}^n\times {\mathbb R}^n\rightarrow {\mathbb R}^n: (\mathbf {x},\mathbf {y})\mapsto \mu (\mathbf {x},\mathbf {y})$. We show that if $\mu$ is a polynomial map in one of the two (sets of) variables $\mathbf {x}$ or $\mathbf {y}$, then $G$ is solvable. Moreover, if one knows that $\mu$ is polynomial in one of the variables, the group $G$ is nilpotent if and only if $\mu$ is polynomial in both its variables.