Solvable Lie Algebras, Lie Groups and Polynomial Structures

Abstract

In this paper, we study polynomial structures by starting on the Lie algebra level, then passing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise, we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotent subalgebras. Using this result, we construct, for any simply connected, connected solvable Lie group G of dim n, a simply transitive action on Rⁿ which is polynomial and of degree ≤ n³. Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group Γ, which is of degree ≤ h(Γ)³ on almost the entire group (h (Γ) being the Hirsch length of Γ).