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We give an $L_\infty$-interpretation of the classification, obtained in[AP2], of the formal deformations of a family of exact Poisson structures indimension three. We indeed obtain again the explicit formulas for all theformal deformations of these Poisson structures, together with a classificationin the generic case, by constructing a suitable quasi-isomorphism between two$L_\infty$-algebras, which are associated to these Poisson structures.

$L_\infty$-interpretation of a classification of deformations of Poisson structures in dimension three