Coalgebraic Approach to the Loday Infinity Category, Stem Differential for 2n-ary Graded and Homotopy Algebras
Annales de l'Institut Fourier, Volume 60 (2010) no. 1, p. 355-387

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 Lod category contains the L 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.

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 Lod contient la catégorie L 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.

DOI : https://doi.org/10.5802/aif.2525
Classification:  16W30,  16E45,  17B56,  17B70
Keywords: Zinbiel coalgebra, graded Loday, Lie, Poisson, Jacobi structure, strongly homotopy algebra, square-zero element method, graded cohomology, Schouten-Nijenhuis, Nijenhuis-Richardson, Grabowski-Marmo bracket, deformation theory
@article{AIF_2010__60_1_355_0,
     author = {Ammar, Mourad and Poncin, Norbert},
     title = {Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {1},
     year = {2010},
     pages = {355-387},
     doi = {10.5802/aif.2525},
     mrnumber = {2664318},
     zbl = {1208.53084},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_1_355_0}
}
Ammar, Mourad; Poncin, Norbert. Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras. Annales de l'Institut Fourier, Volume 60 (2010) no. 1, pp. 355-387. doi : 10.5802/aif.2525. http://www.numdam.org/item/AIF_2010__60_1_355_0/

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