In a paper about Poisson modules that I can't seem to find at the moment, I saw a description of Poisson modules that was rather different from most descriptions I'd seen; however, after considering it, it seemed to naturally extend to many other situations. The description was approximately:

Let $A$ be a Poisson algebra. Then a Poisson module over $A$ is a vector space $E$ and a Poisson algebra structure on $A \oplus E$ such that 1) it agrees with the original structure on $A$, and 2) $\{e_1, e_2\} = e_1 e_2 = 0$ for any $e_1, e_2 \in E$.

The general idea does seem to work for other situations than Poisson; it clearly works for commutative algebras, and works for Lie algebras in the same way as for Poisson algebras.

This leads to a few questions, then. 1) Is this a "good" way to think about algebras (of various forms) and modules? Are there any situations where it doesn't work? 2) The apparent thing that "separates" the "base" algebra structure from the module structure is that the module part is nilpotent (in fact, it "squares" to 0). This made me think back to non-reduced schemes, and made me think that the "right way" to think of a nonreduced scheme is a reduced scheme with extra module structures on top "localized at" the nonreduced points? 3) If so, is there a natural way to think about the "higher modules", since nilpotence can happen at higher powers than 2? 4) Are there any other natural ideas that result from this?

Edited to give a partial answer I thought of afterwards: This really isn't correct; the appropriate idea here is when $E$ is a bimodule. It just happens that in the cases I was thinking of, a module is naturally a bimodule. So if the type of algebra was "arbitrary not-necessarily-commutative algebra", then the above structure would not be equivalent to a module, but a bimodule.


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You might be interested in the Deformation Theoretic content of this. For the commutative algebra version the construction is in S. LICHTENBAUM AND M. SCHLESSINGER, The cotangent complex of a morphism, Trans. Amer. Math. Sot. 128 (1967), 41-70, and shortly afterwards Quillen used this to discuss cohomology in his paper: D. QUILLEN, On the homology of commutative rings, Proc. Sympos. Pure Marh. 17 (1970), 65-87. Quillen ascribes the idea to Beck. Following up the cotangent complex idea takes one through Grothendieck's cofibred categories and Illusie's thesis, which may link up with your later questions.

The idea is also known as Idealisation as it turns a (bi)module over an algebra into an ideal of another algebra.

There is also a very general theory of split epimorphisms in semi-abelian categories (or similar) that describe this process in great generality. (I can give some references for that if needs be, just ask.)

I seem to recall that in non-commutative geometry this construction has been used but that is out of my comfort zone!

  • $\begingroup$ I should mention that the construction is linked to the formation of Kahler differentials in the commutative algebra case, (see the ref to Quillen above.) It also links to Fox derivatives in a Knot theoretic context (see old papers of Crowell.) $\endgroup$
    – Tim Porter
    Sep 26, 2019 at 6:27

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