Journal of Lie Theory

no. 1
On the Structure of Graded Transitive Lie Algebras
Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 265-288
\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 $n\leq 3$.
DOI: 10.5802/jolt.262
Classification: 17B66, 17B70, 17B05
Keywords: Lie algebras, vector fields, graded Lie algebras 
@article{JOLT_2002_12_1_a13,
     author = {G. Post},
     title = {On the {Structure} of {Graded} {Transitive} {Lie} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {265--288},
     year = {2002},
     volume = {12},
     number = {1},
     doi = {10.5802/jolt.262},
     zbl = {1036.17021},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.262/}
}
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G. Post. On the Structure of Graded Transitive Lie Algebras. Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 265-288. doi: 10.5802/jolt.262

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