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Journal of Lie Theory 12 (2002), No. 1, 265--288
Copyright Heldermann Verlag 2002

On the Structure of Graded Transitive Lie Algebras

Gerhard Post
Faculty of Mathematical Sciences, University of Twente, P. O. Box 217, 7500 AE Enschede, The Netherlands


\def\L{{\mathfrak L}} \def\g{{\mathfrak g}} \def\gs{\bar{\g}} We study finite-dimensional Lie algebras $\L$ of polynomial vector fields in $n$ variables that contain the vector fields $\dfrac{\partial}{\partial x_i} \; (i=1,\ldots, n)$ and $x_1\dfrac{\partial}{\partial x_1}+ \dots + x_n\dfrac{\partial}{\partial x_n}$. We show that the maximal ones always contain a semi-simple subalgebra $\gs$, such that $\dfrac{\partial}{\partial x_i}\in \gs \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\gs$-modules in the space spanned by $\dfrac{\partial}{\partial x_i} (i=m+1,\ldots, n)$. The possible algebras $\gs$ are described in detail, as well as all $\gs$-modules that constitute such maximal $\L$. The maximal algebras are described explicitly for $n\leq 3$.

Keywords: Lie algebras, vector fields, graded Lie algebras.

MSC: 17B66; 17B70, 17B05

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