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Journal of Lie Theory 31 (2021), No. 4, 1003--1014
Copyright Heldermann Verlag 2021

Transitive Lie Algebras of Nilpotent Vector Fields and their Tanaka Prolongations

Katarzyna Grabowska
Faculty of Physics, University of Warsaw, Warsaw, Poland

Janusz Grabowski
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

Zohreh Ravanpak
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland


Transitive nilpotent local Lie algebras of vector fields can be easily constructed from dilations $h$ of $\mathbb{R}^n$ with positive weights (give me a sequence of $n$ positive integers and I will give you a transitive nilpotent Lie algebra of vector fields on $\mathbb{R}^n$) as the Lie algebras ${\mathfrak g}_{<0}(h)$ of the polynomial vector fields of negative weights with respect to $h$.\\ We provide a condition for the dilation $h$ such that the Lie algebras of polynomial vectors defined by $h$ are exactly the Tanaka prolongations of the corresponding nilpotent Lie algebras ${\mathfrak g}_{<0}(h)$. However, in some cases of dilations $h$ we can find some `strange' elements of the Tanaka prolongations of ${\mathfrak g}_{<0}(h)$, which we describe in detail. In particular, we give a complete description of derivations of degree $0$ for the Lie algebra ${\mathfrak g}_{<0}(h)$.

Keywords: Vector field, nilpotent Lie algebra, dilation, derivation, homogeneity structures.

MSC: 17B30, 17B66; 57R25, 57S20.

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