# Notion of module over commutative post lie algebra

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 $$[x,y].z = x.(y.z) - y.(x.z)$$ $$x.[y,z] = [x.y,z] + [y,x.z]$$

• 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 $e_1, e_2 \in E$. Sep 23, 2019 at 15:23
• I asked a similar question here: mathoverflow.net/questions/267686/… Sep 23, 2019 at 15:25
• is there a reference for this definition; for my situation the category of those modules must be stable under tensor and Hom Sep 25, 2019 at 17:06
• 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/…) Sep 26, 2019 at 0:27