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Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. What conditions on $\xi\in\frak{g}$ ensure the existence of a (canonical or non-canonical) surjective morphism $\frak{g}\rightarrow\frak{g}_{\xi}$ of complex Lie algebras? Also, what are some of the known decomposition theorems for finite-dimensional complex nilpotent Lie algebras?

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What do you mean by $\mathfrak{g}_\xi$? – MTS Jul 26 '12 at 18:41
I am referring to the stabilizer of $\xi$ under the adjoint represenation of $\frak{g}$. (ie. $\frak{g}_{\xi}=\{\eta\in\frak{g}:[\eta,\xi]=0\}$. – Peter Crooks Jul 26 '12 at 20:34
With no motivation, the first question seems a little senseless to me. For the second question, there is no general decomposition result in the spirit of the decompositions of semisimple Lie algebras, or the decomposition of an arbitrary finite-dimensional Lie algebra (in char 0) as a semidirect product of a semisimple by a solvable one. – YCor Jul 27 '12 at 6:44
Theorem 1.13.19 in Dixmier's "Enveloping algebras" is a decomposition theorem for representations of finite dimensional complex nilpotent Lie algebras. It states that every finite dimensional such representation is uniquely the direct sum of its generalised eigenspaces. – Konstantin Ardakov Jul 27 '12 at 8:04
You can consider that if $\phi:\mathfrak{g}\to\mathfrak{g}_xi$ is an epimorphism of Lie algebras. Then $\mathfrak{g}\mid ker(\phi)\equiv \mathfrak{g}_\xi$. In conclusion $\mathfrak{g}\equiv \mathfrak{g}_\xi\oplus ker(\phi)$ as a vector space, where $ker(\phi)$ is an ideal and $\mathfrak{g}_\xi$ is a subalgebra. Is that the kind of decompositions you are looking for? As a direct sum of an ideal and a subalgebra? – dan232 Jul 27 '12 at 9:06

A well-known decomposition theorem for nilpotent Lie algebras is the weight space decomposition, studied by R. Carles, L. J. Santharoubane and others (in the nilpotent case). Here $L$ is a finite-dimensional nilpotent Lie algebra over an algebraically closed field of characteristic zero, and $Der(L)$ its derivation algebra, $T$ a torus on $L$, that is, a commutative subalgebra of $Der(L)$ consisting of semisimple endomorphisms. This defines naturally a linear representation of $L$, and the elements of $T$ can be diagonalized simultaneously. Therefore we have the decomposition $$ L=\bigoplus_{\alpha \in T^{*}}L^{\alpha}, $$ where $L^{\alpha}$ is the set of $x\in L$ with $t(x)=\alpha (t)x$ for all $t\in T$. The equivalence class of a weight system is an invariant of $L$. It has been studied intensively for nilpotent Lie algebras, e.g., for the classification in low dimensions, and for the cohomology of nilpotent Lie algebras.

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