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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lifts of $C_\infty$- and $L_\infty$-morphisms to $G_\infty$-morphisms


Authors: Grégory Ginot and Gilles Halbout
Journal: Proc. Amer. Math. Soc. 134 (2006), 621-630
MSC (2000): Primary 16E40, 53D55; Secondary 18D50, 16S80
Published electronically: July 18, 2005
MathSciNet review: 2180877
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Abstract: Let $\mathfrak{g}_2$ be the Hochschild complex of cochains on $C^\infty(\mathbb{R}^n)$ and let $\mathfrak{g}_1$ be the space of multivector fields on $\mathbb{R}^n$. In this paper we prove that given any $G_\infty$-structure (i.e. Gerstenhaber algebra up to homotopy structure) on $\mathfrak{g}_2$, and any $C_\infty$-morphism $\varphi$ (i.e. morphism of a commutative, associative algebra up to homotopy) between $\mathfrak{g}_1$ and $\mathfrak{g}_2$, there exists a $G_\infty$-morphism $\Phi$between $\mathfrak{g}_1$ and $\mathfrak{g}_2$ that restricts to $\varphi$. We also show that any $L_\infty$-morphism (i.e. morphism of a 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 $\mathfrak{g}_2$. We also show that any two of such $G_\infty$-morphisms are homotopic.


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Additional Information

Grégory Ginot
Affiliation: Laboratoire Analyse Géométrie et Applications, Université Paris 13 et Ecole Normale Supèrieure de Cachan, France
Email: ginot@cmla.ens-cachan.fr

Gilles Halbout
Affiliation: Institute de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, Strasbourg, France
Email: halbout@math.u-strasbg.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08126-8
PII: S 0002-9939(05)08126-8
Keywords: Deformation quantization, star-product, homotopy formulas, homological methods
Received by editor(s): April 1, 2003
Received by editor(s) in revised form: September 30, 2004
Published electronically: July 18, 2005
Communicated by: Paul Goerss
Article copyright: © Copyright 2005 American Mathematical Society