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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00000272
Contributor : Grégory Ginot <>
Submitted on : Wednesday, June 29, 2005 - 2:41:56 PM
Last modification on : Thursday, January 11, 2018 - 6:12:22 AM
Long-term archiving on : Friday, September 17, 2010 - 5:53:07 PM

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Grégory Ginot, Gilles Halbout. Lift of $C_\infty$ and $L_\infty$ morphisms to $G_\infty$ morphisms. 2005. ⟨hal-00000272v2⟩

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