Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Journal of Lie Theory 12 (2002), No. 1, 265--288Copyright 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 [Abstract-pdf] \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 [ Fulltext-pdf  (258  KB)]