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.
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1 Answer
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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.)
answered Sep 3, 2017 at 16:41
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$\begingroup$ Thanks for your answer. Helped. Can I see any English online article/text ?? $\endgroup$– SKHCommented Sep 3, 2017 at 16:45
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$\begingroup$ Most books on Lie algebras have it online, e.g., here. $\endgroup$ Commented Sep 3, 2017 at 16:54
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$\begingroup$ Thanks but characteristic is not easily found there also . $\endgroup$– SKHCommented Sep 3, 2017 at 18:06
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1$\begingroup$ @SKH your last question (in the comment) is very vague and I can't even guess what you're intending to ask. $\endgroup$– YCorCommented Sep 3, 2017 at 22:02
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2$\begingroup$ 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. $\endgroup$– YCorCommented Sep 7, 2017 at 7:49