It's well known that the $A_{\infty}$ and $L_{\infty}$ operads, being resolutions of the associative and Lie operads, admit descriptions as free operads of certain trees.

The description I am referring to for the $A_{\infty}$ operad is described in Voronov's notes: http://www-users.math.umn.edu/~voronov/8390/lec9.pdf

Similarly the $L_{\infty}$ operad is described in section 2 of the paper by Voronov, Stasheff and Kimura: https://arxiv.org/pdf/hep-th/9307114.pdf

Does the $G_{\infty}$-operad, the free resolution of the Gerstenhaber (Poisson) operad admit a similar description in terms of trees? How can one define it explicitly?


Yes. All of your examples fall within the techniques of Koszul duality theory for operads, and for this you can consult the standard reference of J.-L. Loday and B. Vallette. But also see G. Ginot's thesis http://www.numdam.org/item/AMBP_2004__11_1_95_0/ for explicit descriptions of $\mathsf{Gers}_\infty$ and $\mathsf{Pois}_\infty$.

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