Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators with the Hochschild differential and the Gerstenhaber bracket. A version of the Hochschild-Kostant-Rosenberg theorem shows the cohomology of $D_{poly}^*(A)$ is isomorphic to $\bigoplus_{n\geq -1}\wedge^{n+1}T_X$, polyvector fields. The graded vector space of polyvector fields is also a Lie algebra with the Schouten-Nijenhius bracket. However, the HKR-isomorphism is not a morphism of Lie algebras. The formality theorem of Kontsevich shows the HKR-isomorphism can be corrected to an $L_{\infty}$-morphism whose first term is the HKR-isomorphism. The upshot is this proves star-products on $C^{\infty}(X)$ correspond to formal Poisson structures.
In the article: M. de Wilde and P. B. A Lecomte: An homotopy formula for the Hochschild cohomology, Compositio Mathematica, tome 96, no. 1 (1995), the authors construct an explicit homotopy for the HKR-isomorphism. From this explicit homotopy the authors construct a star-product on $\frak g^*$, the dual of the Lie algebra $\frak g$. My question is the following: can one do the same thing for the general case of a Poisson manifold. In particular, is the explicit homotopy which induces Kontsevich's $L_{\infty}$-quasiisomorphism known? I believe that one has to exist since two $L_{\infty}$-algebras are quasiisomorphic if and only if they are homotopy equivalent. But, can you write down the explicit homotopy from the quasiisomorphism.
Best Answer
The only way I know to construct a formality quasi-morphism for poy-vector fields out of an homotopy is via Tamarkins approach (i.e. $G_\infty$-formality). What Tamrakin does is
prove that there is a suitable $G_\infty$-structure on Hochschild cochains
prove that the obstruction to construct a $G_\infty$-formality step by step is unobstructed. At each step you have to make some choice, and the homotopy for the Hochschild complex of De Wilde-Lecomte gives you a way to do such choices.
But Tamarkin construction (part 1) involves the choice of an associator (i.e. choice of appropriate weights in Kontsevich's $L_\infty$-qausi-isomorphism)... so I think that this is hopeless to construct the formality out of an homotopy.
Moreover, even the other way I don't see how you could associate an homotopy for the Hochschild complex to an $L_\infty$-quasi-isomorphism from $T_{poly}$ to $D_{poly}$.
I might be wrong but I have the feeling that you are mixing two different notions of homotopy: that of higher homotopies in the $L_\infty$-morphism, and that of homotopy retract for the Hochschild cochain complex. In particular, what do you mean by a "homotopy equivalence" between $L_\infty$-algebras (whatever you mean, a homotopy for the Hochschild complex in the sens of De Wilde-Lecomte won't produce an example).