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

Ammar, Mourad; Poncin, Norbert

Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 355-387.

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.

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.

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