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Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to prove is that the tensor product of an $A_\infty$-algebra with a dg-algebra is again an $A_\infty$-algebra.

One can ask similarly whether the tensor product of a $C_\infty$-algebra and an $L_\infty$-algebra is an $L_\infty$-algebra, generalizing the simple fact that the tensor product of a commutative algebra and a Lie algebra is again a Lie algebra. But in fact I can't even prove what should be a much simpler statement: that the tensor product of a $C_\infty$-algebra with a Lie algebra is an $L_\infty$-algebra. Is this true?

Remark. What is quite clear is that the tensor product of a commutative algebra and an $L_\infty$-algebra is an $L_\infty$-algebra.

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If you know the statement for strict algebras, then you "know" the statement for $\infty$-algebras by general nonsense. For example, the fact that a Lie algebra tensored with a commutative algebra again has a canonical Lie structure gives you an operad map $L \to L \otimes C$ (where I use the Boardman--Vogt tensor product, and in this case the map is an iso, but I don't need it to be). I also have surjective quasiisomorphisms $L_\infty \to L$ and $C_\infty \to C$. The tensor product of surjective quasiisomorphisms is again a surjective quasiisomorphism, so I get a surjective quasiisomoprhism $L_\infty \otimes C_\infty \to L \otimes C$. On the other hand, the cofibrancy of $L_\infty$ (in the model in which surjective quasiisomorphisms are acyclic fibrations) implies that I can lift the map $L_\infty \to L \otimes C$ against the map $L_\infty \otimes C_\infty \to L \otimes C$. After doing this lifting, I achieve the map $L_\infty \to L_\infty \otimes C_\infty$. Pulling back along this map gives me an $L_\infty$ structure on the tensor product of $L_\infty$ and $C_\infty$ algberas.

So the question you ask boils down to understanding the map $L_\infty \to L_\infty \otimes C_\infty$. The general nonsense says that it is not strictly canonical, but only canonical up to a contractible space of homotopies. However, it is not too hard to write down an explicit map in terms of "sums over diagrams". Perhaps you are asking for an explicit such sum?

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  • $\begingroup$ Thanks! This only works over a field of characteristic zero, right? If there were a sum over trees formula, one could hope to verify the statement over the integers. $\endgroup$
    – Dan Petersen
    Commented Feb 4, 2018 at 17:42
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    $\begingroup$ When you say the Boardman--Vogt tensor product, do you really mean the Hadamard tensor product? Is there a definition of the BV tensor product for operads in a non-Cartesian monoidal category? $\endgroup$
    – Pavel Safronov
    Commented Feb 4, 2018 at 18:20
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    $\begingroup$ @DanPetersen I think that even $L_\infty$-algebras only work in characteristic 0. $\endgroup$
    – Fernando Muro
    Commented Feb 4, 2018 at 21:57
  • $\begingroup$ @PavelSafronov You know this literature much better than I do, so I trust your namings. I have certainly seen the name "Boardman--Vogt tensor product" used for the operation on dg operads given by $(A\otimes B)(n) = A(n) \otimes B(n)$. This is the monoidal structure that I mean. $\endgroup$
    – Theo Johnson-Freyd
    Commented Feb 5, 2018 at 17:54
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    $\begingroup$ @TheoJohnson-Freyd Do you mind sharing the formula in terms of "sums over diagrams" mentioned at the end of your answer? A reference would also be great! Thanks! $\endgroup$
    – QcH
    Commented Sep 11, 2019 at 15:42

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