You might want to have a look at §2.2 of An $L^\infty$ algebra structure on polyvector fields by Boris Shoikhet, where Boris constructs an exotic $L_\infty$-structure on poly-vector fields on a (possibly infinite dimensional) affine space, which deforms the usual Schouten–Nijenhuis graded Lie structure.

This exotic $L_\infty$-structure has been used in a wise way in An explicit two step quantization of Poisson structures and Lie bialgebras by Merkulov and Willwacher in order to split Kontsevich's quantization procedure into two independent steps (only one of them depending on the choice of an associator).

Furthermore, this exotic $L_\infty$-structure has been shown (by Willwacher in The oriented graph complexes) to be the unique non-trivial $L_\infty$-deformation of the Schouten-Nijenhuis graded Lie structure that is universal (in the sense that it doesn't depend on the dimension $n$ of the affine space one is working with, including $n=+\infty$).

For poly-vector fields on a finite dimensional affine space this exotic $L_\infty$-structure seems to be quasi-isomorphic to the Schouten–Nijenhuis one (Merkulov–Willwacher write in their paper that this is a folklore conjecture; but it seems to be proven in Shoikhet's paper, see the Corollary of the Main Theorem in §2.3).

You might want to have a look at §2.2 of An $L_\infty$ algebra structure on polyvector fields by Boris Shoikhet, where Boris constructs an exotic $L_\infty$-structure on poly-vector fields on a (possibly infinite dimensional) affine space, which deforms the usual Schouten–Nijenhuis graded Lie structure.

This exotic $L_\infty$-structure has been used in a wise way in An explicit two step quantization of Poisson structures and Lie bialgebras by Merkulov and Willwacher in order to split Kontsevich's quantization procedure into two independent steps (only one of them depending on the choice of an associator).

Furthermore, this exotic $L_\infty$-structure has been shown (by Willwacher in The oriented graph complexes) to be the unique non-trivial $L_\infty$-deformation of the Schouten-Nijenhuis graded Lie structure that is universal (in the sense that it doesn't depend on the dimension $n$ of the affine space one is working with, including $n=+\infty$).

For poly-vector fields on a finite dimensional affine space this exotic $L_\infty$-structure seems to be quasi-isomorphic to the Schouten–Nijenhuis one (Merkulov–Willwacher write in their paper that this is a folklore conjecture; but it seems to be proven in Shoikhet's paper, see the Corollary of the Main Theorem in §2.3).

You might want to have a look at $\S2.2$ of this paper by Boris Shoikhet, where Boris constructs an exotic $L_\infty$-structure on poly-vector fields on a (possibly infinite dimensional) affine space, which deforms the usual Schouten-Nijenhuis graded Lie structure.

This exotic $L_\infty$-structure has been used in a wise way in another paper by Merkulov and Willwacher in order to split Kontsevich's quantization procedure into two independant steps (only one of them depending on the choice of an associator).

Furthermore, this exotic $L_\infty$-structure has been shown (by Willwacher) to be the unique non-trivial $L_\infty$-deformation of the Schouten-Nijenhuis graded Lie structure that is universal (in the sense that it doesn't depend on the dimension $n$ of the affine space one is working with, including $n=+\infty$).

For poly-vector fields on a finite dimensional affine space this exotic $L_\infty$-structure seems to be quasi-isomorphic to the Schouten-Nijenhuis one (Merkulov-Willwacher write in their paper that this is a folklore conjecture; but it seems to be proven in Shoikhet's paper, see the Corollary of the Main Theorem in $\S2.3$).

You might want to have a look at §2.2 of An $L^\infty$ algebra structure on polyvector fields by Boris Shoikhet, where Boris constructs an exotic $L_\infty$-structure on poly-vector fields on a (possibly infinite dimensional) affine space, which deforms the usual Schouten–Nijenhuis graded Lie structure.

This exotic $L_\infty$-structure has been used in a wise way in An explicit two step quantization of Poisson structures and Lie bialgebras by Merkulov and Willwacher in order to split Kontsevich's quantization procedure into two independent steps (only one of them depending on the choice of an associator).

Furthermore, this exotic $L_\infty$-structure has been shown (by Willwacher in The oriented graph complexes) to be the unique non-trivial $L_\infty$-deformation of the Schouten-Nijenhuis graded Lie structure that is universal (in the sense that it doesn't depend on the dimension $n$ of the affine space one is working with, including $n=+\infty$).

For poly-vector fields on a finite dimensional affine space this exotic $L_\infty$-structure seems to be quasi-isomorphic to the Schouten–Nijenhuis one (Merkulov–Willwacher write in their paper that this is a folklore conjecture; but it seems to be proven in Shoikhet's paper, see the Corollary of the Main Theorem in §2.3).