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Let $ (S, \{. \}, [.]) $ be an algebra over a vector space endowed with two bilinear maps $ \{. \}, [.] : S \times S \rightarrow S$ and satisfying some compatibility conditions.

In example, if S is a Poisson algebra; the structure of Poisson module over S is well known

im asking if there is ( in a abstract way) a general definition of module over $S$ compatible with the two structures $\{. \}, [.] $

Edited; the example i am interested in, is when $S$ is a CPA-algebra (link); that is $ (S, [.]) $ is a Lie algebra and $ (S,.) $ a commutative algebra such that $$[x,y].z = x.(y.z) - y.(x.z)$$ $$x.[y,z] = [x.y,z] + [y,x.z]$$

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    $\begingroup$ In some sense, the Poisson module can be better thought of as a Poisson bimodule because of the antisymmetry of the bracket and symmetry of multiplication. If you accept that, then there is a relatively natural definition: a module $E$ is a vector space (over the same underlying field) such that $S \oplus E$ is the "same kind" of algebra (i.e. it satisfies the same compatibility conditions), where you set $\{e_1, e_2\} = [e_1, e_2] = 0$ for $e_1, e_2 \in E$. $\endgroup$
    – user44191
    Sep 23, 2019 at 15:23
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    $\begingroup$ I asked a similar question here: mathoverflow.net/questions/267686/… $\endgroup$
    – user44191
    Sep 23, 2019 at 15:25
  • $\begingroup$ is there a reference for this definition; for my situation the category of those modules must be stable under tensor and Hom $\endgroup$
    – Moutand Mohammed
    Sep 25, 2019 at 17:06
  • $\begingroup$ I haven't really worked with the definition, so I can only give you the answers given me; darij grinberg's comment uses Triples, Algebras, and Cohomology (tac.mta.ca/tac/reprints/articles/2/tr2.pdf), while another comment gives nLab(ncatlab.org/nlab/show/…) $\endgroup$
    – user44191
    Sep 26, 2019 at 0:27

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I think you are asking for the notion of a module over an algebra over an operad. In your case, the operad is the Poisson operad.

You can find a brief reference in nLab. And more details in Loday-Vallette's book "Algebraic Operads"; particularly in section 12.3.1. Even more details in this paper by Berger and Moerdijk: arXiv.

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