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When discussing Levi decomposition wikipedia mentions real finite dimensional Lie algebras and later says such decomposition is not available in infinite dimension and in positive characteristic. From other sources I came to know such decomposition is available over $\mathbb{R} $ and $\mathbb{C}$.What I wonder that whether it is true over all $char=0$ fields ? Any help(may be with some reference) is appreciated. Thanks.

SKH
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Yes, the theorem of Levi-Malcev holds for finite dimensional Lie algebras over any field of characteristic 0. See Bourbaki, Groupes et Algebres de Lie, ch. I, §6 no. 8. (Théorème 5 in particular.)

Armando j18eos
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Torsten Schoeneberg
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  • Thanks for your answer. Helped. Can I see any English online article/text ?? – SKH Sep 03 '17 at 16:45
  • Most books on Lie algebras have it online, e.g., here. – Dietrich Burde Sep 03 '17 at 16:54
  • Thanks but characteristic is not easily found there also . – SKH Sep 03 '17 at 18:06
  • And also there are many possible semidirect products . It is not clear which particular semidirect product the theorem talks about . I hope it is when the action of semisimple part on the radical is given by adjoints .But then if we consider the Lie algebra to be one from the other semidirect products ,then what will be the Levi decomposition for that ? – SKH Sep 03 '17 at 18:06
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    @SKH your last question (in the comment) is very vague and I can't even guess what you're intending to ask. – YCor Sep 03 '17 at 22:02
  • What I know about semidirect product(s) of $\mathfrak{g}$ and $\mathfrak{h} $ where $\mathfrak{g}$ is a subalgebra and $\mathfrak{h}$ is an ideal is that it corresponds to one map $\phi :\mathfrak{g} \rightarrow \mathfrak{der h}$. And Levi decomposition says any finite dimensional Lie algebra(char=0) is the semidirect product of ... . What I wanted to ask is that the semidirect product corresponds to which $\phi$? Is it assumed that the semidirect product is corresponding to $\phi(x)= ad x|_{\mathfrak{h}}$ ? – SKH Sep 05 '17 at 07:09
  • If so then how to write other semidirect products (corresponding to some other $\phi $'s) in that form corresponding to the above $\phi$ ? In particular, is $\mathfrak{g} \ltimes \psi \mathfrak{h} = \mathfrak{g} \ltimes \mathfrak{h}$ is the Levi decomposition for any $\psi \in Hom{Lie} (\mathfrak{g} , \mathfrak{ der h})$ ? – SKH Sep 05 '17 at 07:23
  • It should not be the case I guess ! – SKH Sep 05 '17 at 07:29
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    Of course when you consider a Levi decomposition (or any semidirect decomposition) of a Lie algebra, ie decompose the Lie algebra as direct sum of an ideal and a subalgebra, you consider the adjoint action of the splitting subalgebra, not any exotic other action. There's no ambiguity about this in this context. – YCor Sep 07 '17 at 07:49