Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-05-08126-8
Published electronically: July 18, 2005
MathSciNet review: 2180877
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [BFFLS1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Quantum mechanics as a deformation of classical mechanics, Lett. Math. Phys. 1 (1975), 521-530. MR 0674337 (58:32617)
  • [BFFLS2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization, I and II, Ann. Phys. 111 (1977), 61-151. MR 496157 (58:14737a); MR 0496158 (58:14737b)
  • [CFT] A. S. Cattaneo, G. Felder, L. Tomassini, From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115 (2002), 329-352.MR 1944574 (2004a:53114)
  • [Gi] G. Ginot, Homologie et modèle minimal des algèbres de Gerstenhaber, Ann. Math. Blaise Pascal 11 (2004), 91-127.MR 2077240
  • [GH] G. Ginot, G. Halbout, A formality theorem for Poisson manifold, Lett. Math. Phys. 66 (2003), 37-64. MR 2064591
  • [GK] V. Ginzburg, M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), 203-272. MR 1301191 (96a:18004)
  • [Ha] G. Halbout, Formule d'homotopie entre les complexes de Hochschild et de de Rham, Compositio Math. 126 (2001), 123-145. MR 1827641 (2002d:16010)
  • [Hi] V. Hinich, Tamarkin's proof of Kontsevich's formality theorem, Forum Math. 15 (2003), 591-614. MR 1978336 (2004j:17029)
  • [HKR] G. Hochschild, B. Kostant and A. Rosenberg, Differential forms on regular affine algebras, Transactions AMS 102 (1962), 383-408.MR 0142598 (26:167)
  • [Ka] C. Kassel, Homologie cyclique, caractère de Chern et lemme de perturbation, J. Reine Angew. Math. 408 (1990), 159-180. MR 1058987 (92d:19002)
  • [Ko1] M. Kontsevich, Formality conjecture. Deformation theory and symplectic geometry, Math. Phys. Stud. 20 (1996), 139-156. MR 1480721 (98m:58044)
  • [Ko2] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216. MR 2062626
  • [KS] M. Kontsevich, Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, Math. Phys. Stud. 21 (2000), 255-307.MR 1805894 (2002e:18012)
  • [Kos] J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie in ``Elie Cartan et les mathémati-ques d'aujourd'hui", Astérisque (1985), 257-271. MR 0837203 (88m:17013)
  • [LS] T. Lada, J. D. Stasheff, Introduction to SH Lie algebras for physicists, Internat. J. Theoret. Phys. 32 (1993), 1087-1103. MR 1235010 (94g:17059)
  • [Ta] D. Tamarkin, Another proof of M. Kontsevich's formality theorem, math. QA/9803025.
  • [TS] D. Tamarkin, B. Tsygan, Noncommutative differential calculus, homotopy BV algebras and formality conjectures, Methods Funct. Anal. Topology 6 (2000), 85-97. MR 1783778 (2001i:16017)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16E40, 53D55, 18D50, 16S80

Retrieve articles in all journals with MSC (2000): 16E40, 53D55, 18D50, 16S80


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: https://doi.org/10.1090/S0002-9939-05-08126-8
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

American Mathematical Society