Abstract
In this paper, we study polynomial structures by starting on the Lie algebra level, thenpassing 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 nilpotentsubalgebras. Using this result, we construct, for any simply connected, connected solvable Lie groupG of dim n, a simply transitive action on R n 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).
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Dekimpe, K. Solvable Lie Algebras, Lie Groups andPolynomial Structures. Compositio Mathematica 121, 183–204 (2000). https://doi.org/10.1023/A:1001738932743
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DOI: https://doi.org/10.1023/A:1001738932743