On deformation theory 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 Moyal’s 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.