What about Lie algebras over commutative rings?

It is just like the associative algebra over commutative ring (advanced linear algebra). It is a natural extension and can make the structure of Lie algebra more algebraic, but I find little book discussing this topic. Does somebody know something of this?

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2 Answers

There is a algebra named Lie–Rinehart algebra which is a pair consisting of a commutative algebra A and a Lie algebra g such that A is a g-module and g is an A-module with both module structures being compatible in the way: $\theta:g\to Der(A)$ is a Lie algebra morphism from g to the derived Lie algebra of A, $[x,ay]=\theta(x)(a)y+a[x,y]$ where $x, y\in g, a\in A$. It coming form Poisson geometry and have many beatiful algebraic properties. See the articles by Johannes Huebschmann on arxiv, for example http://arxiv.org/abs/math/0303016

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This is more a comment than an answer- I cannot "comment" so I put it here:

Weibels Book discusses Lie Algebra cohomology for Lie algebras over commutative rings if I am not mistaken.

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Yes，but that is just a form,do not pay more attention to the Lie algebra and show some difference between field coefficient and ring coefficient. –  Strongart Dec 25 '11 at 6:13