The explicit equivalence between the standard and the logarithmic star product for Lie algebras. I. (Q456622)

This paper gives an explicit proof of the equivalence between the standard and the logarithmic star product on the symmetric algebra of a Lie algebra. Let us recall that Kontsevich's formality theorem stipulates the existence of a star product on any Poisson manifold. Kontsevich's work enables also to endow the symmetric algebra \(S(g)\) of a Lie algebra \(g\) with a star-product \(\ast\) and to obtain an isomorphism of associative algebras between \((S(g),\ast)\) and \((U(g),.)\). In this article, the author establishes an explicit equivalence between \((S(g),\ast_{\log})\) and \((U(g),.)\), where \(\ast_{\log}\) is the logarithmic star product. This result is a consequence of the logarithmic version of the formality theorem. The link between the equivalence result and the Grothendieck-Teichmüller group is also pointed out.