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In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) the following definition is given:

A Lie structure over the $R$-module $M$ is a $T(M)$-bimodule $A$ together with a bilinear function $M\otimes M\to A$ taking $x \otimes y \mapsto \langle x,y\rangle$, satisfying

  • $\langle x,x\rangle = 0$;

  • $\langle x,y\rangle t(uv-vu)=(xy-yx)t\langle u,v\rangle$, for all $x,y,u,v \in M$ and $t \in T(M)$; and

  • $(\langle x,y\rangle z-z\langle x,y\rangle)+(\langle y,z\rangle x-x\langle y,z\rangle)+(\langle z,x\rangle y-y\langle z,x\rangle) = 0$, for $x,y,z \in M$.

My question is, how does this generalize the case of a Lie algebra over a field? And what is the motivation behind the second condition? Why can't we simply define a Lie structure over a ring to be an alternating bilinear law which satisfies the Jacobi identity?

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    $\begingroup$ At the end, you mean an alternating bilinear law, not alternating linear form $\endgroup$
    – YCor
    Commented Mar 24, 2019 at 13:16
  • $\begingroup$ I don't understand. What's the difference between form and law? $\endgroup$
    – nobody
    Commented Mar 24, 2019 at 14:40
  • $\begingroup$ A form is scalar-valued $\endgroup$
    – YCor
    Commented Mar 24, 2019 at 15:41
  • $\begingroup$ Edited, thanks @YCor $\endgroup$
    – nobody
    Commented Mar 25, 2019 at 15:58
  • $\begingroup$ Anyway what you say at the end is also what would be naturally called Lie structure on $M$. I don't see the motivation for the definition in the paper you're quoting, and I don't know if this definition has been used in subsequent papers, with this terminology or with another one. You might get information using Google Scholar or MathSciNet, which indicates papers quoting this 1969 paper (MSN gives only 7 quotations, but it usually only detects those whose year is approximately $\ge 2000$). $\endgroup$
    – YCor
    Commented Mar 25, 2019 at 16:16

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