On the Obstruction of the Deformation Theory in the DGLA of Graded Derivations

Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 21)

Abstract

In a recent paper, the authors studied the deformation theory in the DGLA of graded derivations \(\mathcal{D}^{{\ast}}\left (M\right )\) of differential forms on M. They proved the existence of canonical solutions \(e_{\Phi }\) of Maurer-Cartan equation depending on a vector valued differential form \(\Phi\) and gave a classification of these canonical solutions by their type: a canonical solution \(e_{\Phi }\) is of finite type r if \(\Phi ^{r}\left [\Phi,\Phi \right ]_{\mathcal{F}\mathcal{N}} = 0\) and \(r =\min \left \{\,j \in \mathbb{N}: \Phi ^{\,j}\left [\Phi,\Phi \right ]_{\mathcal{F}\mathcal{N}} = 0\ \right \}\), where \(\left [\cdot,\cdot \right ]_{\mathcal{F}\mathcal{N}}\) is the Frölicher-Nijenhuis bracket. In this paper it is shown that the deformation theory in the DGLA of graded derivations is not obstructed, but it is level-wise obstructed.

Keywords

Differential graded Lie algebras Graded derivations Maurer-Cartan equation 

References

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Copyright information

© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università degli Studi di FirenzeFirenzeItaly
  2. 2.Institut de Mathématiques, UMR 7586 du CNRS, Case 247Université Pierre et Marie-CurieParis Cedex 05France

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