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H 1 Deformations of $N$-differential graded algebras

Auteurs : Díaz, Rafael (Auteur de la Conférence)
CIRM (Editeur )

Résumé : We introduce the concept of N-differential graded algebras ($N$-dga), and study the moduli space of deformations of the differential of a $N$-dga. We prove that it is controlled by what we call the $N$-Maurer-Cartan equation. We provide geometric examples such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives. We also consider deformations of the differential of a $q$-differential graded algebra. We prove that it is controlled by a generalized Maurer-Cartan equation. We find explicit formulae for the coefficients involved in that equation. Deformations of the $3$-differential of $3$-differential graded algebras are controlled by the $(3,N)$ Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of $N$-differential graded algebras, and use these results to study $N$-Lie algebroids. We study higher depth algebras, and work towards the construction of the concept of $A^N_ \infty$-algebras.

Codes MSC :
16E45 - Differential graded algebras and applications
53B50 - Applications to physics
58B32 - Geometry of quantum groups
81R10 - Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, W-algebras and other current algebras and their representations

 Informations sur la Vidéo Langue : Anglais Date de publication : 11/12/14 Date de captation : 04/12/14 Collection : Research talks ; Algebra ; Algebraic and Complex Geometry ; Mathematical Physics Format : quicktime ; audio/x-aac Durée : 00:30:56 Domaine : Algebra ; Mathematical Physics ; Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2014-12-04_Diaz.mp4 Informations sur la rencontre Nom de la rencontre : Algebra, deformations and quantum groups / Algèbre, déformations et groupes quantiquesOrganisateurs de la rencontre : Kulish, Peter ; Makhlouf, Abdenacer ; Paal, Eugen ; Schlichenmaier, Martin ; Stolin, AlexanderDates : 01/12/14 - 05/12/14 Année de la rencontre : 2014 Citation Data DOI : 10.24350/CIRM.V.18647103 Cite this video as: Díaz, Rafael (2014). Deformations of $N$-differential graded algebras. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18647103 URI : http://dx.doi.org/10.24350/CIRM.V.18647103

Bibliographie

1. Angel, M., & Díaz, R. (2007). On $N$-differential graded algebras. Journal of Pure and Applied Algebra, 210(3), 673-683 - http://dx.doi.org/10.1016/j.jpaa.2006.11.009

2. Angel, M., & Díaz, R. (2007). $N$-flat connections. In S. Paycha, & B. Uribe (Eds.), Geometric and topological methods for quantum field theory (pp. 163-172). Providence, RI: American Mathematical Society (AMS) - http://dx.doi.org/10.1090/conm/434/08345

3. Angel, M., Camacaro, J., & Díaz, R. (2007). On the $(3, N)$ Maurer-Cartan equation. Journal of Nonlinear Mathematical Physics, 14(4), 548-569 - http://dx.doi.org/10.1080/jnmp.2007.14.4.4

4. Angel, M., & Díaz, R. (2006) $A^N_\infty$-algebras. Preprint, arXiv:math/0612661v1 [math.QA] - http://arxiv.org/abs/math/0612661

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