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

Abstract

We define a graded twisted-coassociative coproduct on the tensor algebra TW of anyZ n -graded vector space W . If W is the desuspension space # V of a graded vector space V , the coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1to-1 with sequences p s, s ‚ 1, of s-linear maps on V (resp. Z n -graded Loday structures on V , sequences that we call Loday infinity structures on V ). We prove a minimal model theorem for Loday infinity algebras, investigate Loday infinity morphisms, 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 (the latter extend the corresponding Lie algebras in the sense of Michor and Vinogradov). These algebraic structures have square zero with respect to the stem bracket, so that we obtain natural cohomological theories that have good properties with respect to formal deformations. The stem bracket restricts to the graded Nijenhuis-Richardson and— up to isomorphism—to the Grabowski-Marmo brackets (the last bracket extends the SchoutenNijenhuis bracket to the space of graded antisymmetric first order polydifferential operators), 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.