Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an L-infinity-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.
Graph complexes in deformation quantization / Fiorenza, Domenico; Lucian M., Ionescu. - In: LETTERS IN MATHEMATICAL PHYSICS. - ISSN 0377-9017. - 73:3(2005), pp. 193-208. [10.1007/s11005-005-0017-7]