For my thesis, I'm trying to put the structure of an $L_\infty$-algebra on a tensor product $A\otimes L$ of a $L_\infty$-algebra $L$ with a differential, graded commutative algebra $A$. I know that there are arguments by abstract homotopical nonsense to prove the existence of such a structure, but I'm looking for a more explicit approach.

My idea is to define the higher brackets as $$\tilde l_n(\alpha_1\otimes v_1,\cdots,\alpha_n\otimes v_n)=\eta \alpha_1\cdots\alpha_n l_n(v_1,\cdots,v_n)$$ while the $1$-brackets are $\tilde l_1(\alpha\otimes v)=d\alpha\otimes v+(-1)^{|\alpha|}\alpha\otimes l_1(v)$. Here $\eta=\eta(\alpha_1\otimes v_1,\cdots,\alpha_n\otimes v_n)=(-1)^{\sum_{1\leq i<j\leq n}|v_i||\alpha_j|}$, so that, for example, $\eta(\alpha_1\otimes v_1,\alpha_2\otimes v_2)=(-1)^{|v_1||\alpha_2|}$.

My question is if this is something already in the literature (I know it is the way brackets and differentials are defined when $L$ is a dg Lie algebra).

  • 6
    $\begingroup$ You can see this discussed, e.g., in Loday-Vallette, in the version on Vallette's website (math.univ-paris13.fr/~vallette/Operads.html, section numbering may differ slightly from the published version), Section 5.3.3 (Hadamard product of operads) and proposition 5.3.4 along with Appendix B.3 (as mentioned there) for discussion of the signs. I didn't double check your signs but the formulas you give here appear to be a direct application of 5.3.4 to $Com$ and $L_\infty$, along with the identification at the level of operads of $Com\otimes L_\infty$ with $L_\infty$. $\endgroup$ May 26 at 6:05
  • 3
    $\begingroup$ Thank you very much. $\endgroup$ May 26 at 6:11


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