Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras

[Approche coalgébrique de la catégorie des algèbres de Loday infinies, différentielle souche pour les algèbres graduées $2n$-aires ou à  homotopie]

Ammar, Mourad ; Poncin, Norbert

Annales de l'Institut Fourier, Tome 60 (2010), p. 355-387 / Harvested from Numdam

Résumé

Nous définissons un coproduit gradué et coassociatif tordu sur l’algèbre tensorielle d’un espace vectoriel gradué $V$. Les codérivations (resp. codifférentielles quadratiques de “degré 1”, codifférentielles impaires quelconques) de cette co-algèbre sont en correspondance biunivoque avec les suites d’applications multilinéaires sur $V$ (resp. structures graduées de Loday sur $V$, suites que nous appelons structures de Loday infinies sur $V$). Nous prouvons un théorème du modèle minimal pour les algèbres infinies de Loday et observons que la catégorie $\text{Lod}_{\!\infty }$ contient la catégorie $\text{L}_{\!\infty }$ comme sous-catégorie. En plus, le crochet de Lie gradué des codérivations conduit à un crochet de Lie gradué “souche” sur les espaces des cochaînes des algèbres de Loday graduées, de Loday infinies et de Loday graduées $2n$-aires. Le crochet souche se restreint aux crochets gradués de Nijenhuis-Richardson et de Grabowski-Marmo, et il encode, au-delà  des cohomologies déjà  mentionnées, celles des algèbres de Lie graduées, de Poisson graduées, de Jacobi graduées, Lie infinies, ainsi que celle des algèbres de Lie graduées $2n$-aires.

Abstract

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space $V$. The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on $V$ (resp. graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras and observe that the $\text{Lod}_{\!\infty }$ category contains the $\text{L}_{\!\infty }$ category as a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie “stem” bracket on the cochain spaces of graded Loday, Loday infinity, and $2n$-ary graded Loday algebras. This stem bracket restricts to the graded Nijenhuis-Richardson and Grabowski-Marmo brackets, and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of $2n$-ary graded Lie algebras.

Détail

Publié le : 2010-01-01

MR 2664318 | Zbl 1208.53084

DOI : https://doi.org/10.5802/aif.2525
Classification:  16W30,  16E45,  17B56,  17B70
Mots clés: co-algèbre de Zinbiel, suites graduées de Loday, Lie, Poisson, structure de Jacobi, algèbre fortement homotopique, cohomologie, Schouten-Nijenhuis, Nijenhuis-Richardson, crochets gradués de Grabowski-Marmo, théorie de déformation

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