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Let $g$ be a Lie ring (Lie algebra over $\mathbb Z$), and let $U(g)$ and $S(g)$ denote the universal enveloping and symmetric algebra of $g$. The Poincaré-Birkhoff-Witt theorem (in the form proved by Lazard, see "Sur les algebres enveloppantes universelles de certaines algebres de Lie, M Lazard - Publ. Sci. Univ. Alger. Ser. A, 1954) yields a ring isomorphism between $S(g)$ and an associated graded of $U(g)$.

I can prove that $S(g)$ and $U(g)$ are isomorphic as $\mathbb Z$-modules; this essentially follows from the proof by Lazard. Is this already known?

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  • $\begingroup$ Just remark isomorphism fails for char = p mathoverflow.net/questions/99018/… $\endgroup$ Aug 6 '12 at 10:43
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    $\begingroup$ Doc, you make my head spin. Surely, PBW holds over any field. $\endgroup$
    – Bugs Bunny
    Aug 7 '12 at 12:53
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    $\begingroup$ Clarification: I consider algebras over $\mathbb Z$, in which case Lazard shows that there is a ring isomorphism between $gr(U(g))$ and $S(g)$. Indeed $\mathbb Z$ is a PID. The typical "counterexamples" to PBW involve algebras over non-domains such as $\mathbb F_p[a,b,c]/(a^p,b^p,c^p)$. My question is about isomorphism of $S(g)$ and $U(g)$ as abelian groups, not as rings. $\endgroup$
    – grok
    Aug 8 '12 at 11:29
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Earlier Lazard's result (Sur les algèbres enveloppantes universelles de certaines algèbres de Lie. C. R. Acad. Sci. Paris 234, (1952). 788–791. ) does it for Lie rings over PID.

For general Lie rings the first counterexample was constructed by Shirshov (On the representation of Lie rings as associative rings. Uspehi Matem. Nauk (N.S.) 8, (1953). no. 5(57), 173–175.)

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It is known if $g$ is free as a $\mathbb{Z}$-module, and fails otherwise, see e.g. P.M.Cohn, A Remark on the Birkhoff-Witt Theorem J. London Math. Soc. (1963) s1-38(1): 197-203

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    $\begingroup$ I don't think this is claimed anywhere in Cohn. $\endgroup$ Aug 6 '12 at 15:02
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    $\begingroup$ It fails in the category of "algebras over $\Phi$", for some rings $\Phi$ that contain zero divisors; however, this is not exactly the question I wanted to ask; I added a clarification. $\endgroup$
    – grok
    Aug 8 '12 at 11:30
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    $\begingroup$ @Dotsenko: yes indeed. Most people seek an algebra isomorphism, so replace $U(g)$ by an associated graded; I'm interested in less, but don't want to pass to an associated graded. My question only makes sense for Lie rings with $\mathbb Z$ or $\mathbb Z/q$ additive factors with $q$ non-prime. $\endgroup$
    – grok
    Aug 9 '12 at 12:03
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    $\begingroup$ Does somebody have the paper by by Lazard ? (I cannot find it) Sur les algebres enveloppantes universelles de certaines algebres de Lie, M Lazard - Publ. Sci. Univ. Alger. Ser. A, 1954 $\endgroup$ Feb 14 '13 at 14:50
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    $\begingroup$ Paper by Cohn: justpasha.org/math/links/files/c/cohn/197.pdf $\endgroup$ May 10 '13 at 15:38

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