Lifts of $L_\infty$-morphims to $G_\infty$-morphisms

Grégory Ginot (1) , Gilles Halbout (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let $\g_2$ be theHochschild complex of cochains on $C^\infty(\RM^n)$ and $\g_1$ be the space of multivector fields on $\RM^n$.In this paper we prove that given any $G_\infty$-structure ({\rm i.e.}Gerstenhaber algebra up to homotopy structure) on $\g_2$,and any morphism $\varphi$ of Lie algebra up to homotopy between$\g_1$ and $\g_2$, there exists a $G_\infty$-morphism $\Phi$between$\g_1$ and $\g_2$ that restricts to $\varphi$. In particular, themorphism constructed by Kontsevich can be obtained using Tamarkin'smethod for any $G_\infty$-structure on $\g_2$. We also show thatany two of such $G_\infty$-morphisms are homotopic in a certain sense.

Mots clés: Deformation quantization; star-product; homotopy formulas; homological methods

Domaines: Algèbres quantiques [math.QA]; Anneaux et algèbres [math.RA]; Géométrie différentielle [math.DG]

Grégory Ginot, Gilles Halbout. Lifts of $L_\infty$-morphims to $G_\infty$-morphisms. 2003. ⟨hal-00000272v1⟩