On the Structure of Graded Transitive Lie Algebras

G. Post

Journal of Lie theory, Tome 12 (2002) no. 1, pp. 265-288.

Résumé

\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 $n\leq 3$.

Détail

Classification : 17B66, 17B70, 17B05
Mots-clés : Lie algebras, vector fields, graded Lie algebras