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I see in Wikipeida that every Lie algebra is either constructed from an associative algebra by defining: $[x,y]=xy-yx$, or a subalgebra of a Lie algebra thus constructed. Where can I find a proof?

Moreover, is there any existing example of a Lie algebra which cannot from constructed in this way be an associative algebra?

Thanks a lot :)

Fie
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  • The same wiki page you are referring to lists many examples of Lie algebras that do not come from an associative algebra. It doesn't list free Lie algebras which is a very big source of Lie algebras not coming from associative algebras. – Ittay Weiss Nov 06 '12 at 04:13
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    Search for universal enveloping algebra. It is a theorem that the the natural map from Lie algebra to its UEA is injective (follows from PBW theorem), which means that it is a subalgebra of a Lie algebra that comes from an associative algebra. –  Nov 06 '12 at 04:14
  • my apologies, I missed the 'or a subalgebra...'. – Ittay Weiss Nov 06 '12 at 07:28
  • See http://math.stackexchange.com/questions/3031/cayleys-theorem-for-lie-algebras – David E Speyer Nov 06 '12 at 15:30

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