Abstract: We study finite-dimensional Lie algebras of polynomial vector fields in $n$ variables that contain all partial derivatives and the Euler operator. We derive some general results on the structure of such Lie algebras, and provide the complete classification in the cases $n=2$ and $n=3$. Finally we describe a certain construction in high dimensions.
Classification (MSC2000): 17B66, 17B70, 17B05
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