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
konieczn@fuw.edu.pl
Janusz Grabowski
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
jagrab@impan.pl
Zohreh Ravanpak
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
zravanpak@impan.pl
[Abstract-pdf]
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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