A canonical semi-classical star product

We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controlling the deformation theory of associative algebras. We prove that there is a canonical solution (choice independent) within the class of graphs without circuits, i.e. at the level of the free operad, without imposing the Jacobi identity.

The proof is a consequence of the unique factorization property of the pre-Lie algebra of graphs (tree operad), where composition is the insertion of graphs. The restriction to graphs without circuits, i.e. at “tree level”, accounts for the interpretation as a semi-classical solution. The fact that this solution is canonical should not be surprising, in view of the Hausdorff series, which lies at the core of almost all quantization prescriptions.

Invitation to operadic dynamics

In 1963, Gerstenhaber invented an operad calculus in the Hochschild complex of an associative algebra; operads were introduced under the name of pre-Lie systems. In the same year, Stasheff constructed quite an original geometrical operad, which nowadays is called an associahedra. The notion of an operad was further formalised by May as a tool for iterated loop spaces. The main principles of the operad calculus (brace algebra) were presented by Gerstenhaber and Voronov. Some quite remarkable research activity in the operad theory and its applications can be observed in the last decade. It may be said that operads are also becoming an important tool for Quantum Field Theory and deformation quantization.

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