Lift of $C_\infty$ and $L_\infty$ morphisms to $G_\infty$ morphisms

Grégory Ginot (1) , Gilles Halbout (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let $\g_2$ be the Hochschild 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 $C_\infty$-morphism $\varphi$ ({\rm i.e.} morphism of commutative, associative 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 $\varphi$. We also show that any $L_\infty$-morphism ({\rm i.e.} morphism of Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a $G_\infty$-morphism, using Tamarkin's method for any $G_\infty$-structure on $\g_2$. We also show that any two of such $G_\infty$-morphisms are homotopic.

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. Lift of $C_\infty$ and $L_\infty$ morphisms to $G_\infty$ morphisms. Proceedings of the American Mathematical Society, 2006, 134, pp.621-630. ⟨10.1090/S0002-9939-05-08126-8⟩. ⟨hal-00000272v3⟩