I must prove that if $L$ is a Lie algebra and denoting $U(L)$ the enveloping algebra, then $U(L)$ hasn't zero divisions (e.g. if $ab=0 \,\,\, a,b \in U(L)$ then $a=0$ or $b=0$). Some ideas?
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hint: use the PBW basis. – Matthew Towers Feb 23 '13 at 11:23
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@mt_ What is it? – Raffaele Dolce Feb 23 '13 at 11:50
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Have a look at http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem -- the PBW basis is the basis $x^ay^b\cdots $ where $x,y,\ldots$ is a basis of the Lie algebra. The PBW basis shows that the enveloping algebra has a filtration whose associated graded algebra is a polynomial algebra, so is a domain. – Matthew Towers Feb 23 '13 at 12:40
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@mt_ Could you give me a sckhet of proof? – ArthurStuart Feb 23 '13 at 22:56
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Here is a rough sketch. 1) the UEA has a filtration such that the associated graded algebra is isomorphic to the polynomial algebra on the Lie algebra (this is the content of the PBW theorem) 2) if a ring $R$ has a filtration whose associated graded is a domain, then $R$ is a domain. This is in Jacobson's book Lie Algebras for example – Matthew Towers Feb 24 '13 at 13:19