On the Structure of Graded Transitive Lie Algebras

Gerhard Post

Abstract: We study finite-dimensional Lie algebras $\L$ of polynomial vector fields in $n$ variables that contain the vector fields ${{\partial}\over{\partial x_i}} \; (i=1,\ldots, n)$ and $x_1{{\partial}\over{\partial x_1}}+ \dots + x_n{{\partial}\over{\partial x_n}}$. We show that the maximal ones always contain a semi-simple subalgebra $\gs$, such that ${{\partial}\over{\partial x_i}}\in \gs \; (i=1,\ldots, m)$ for an $m$ with $1 \leq m \leq n$. Moreover a maximal algebra has no trivial $\gs$-modules in the space spanned by ${{\partial}\over{\partial x_i}} (i=m+1,\ldots, n)$. 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

Classification (MSC2000): 17B66, 17B70, 17B05

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