On deformation of associative algebras and graph homology

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an “almost contraction” leading to a canonical solution of the corresponding Maurer–Cartan equation is noted. This role is reminiscent of the Homotopical Perturbation Lemma, with the infinitesimal deformation cocycle as “initiator.” Applied to star-products, we show how Moyalʹs formula can be obtained using such an almost contraction and conjecture that the “merger operation” provides a canonical solution at least in the case of linear Poisson structures.