We define a graded twisted-coassociative coproduct on the tensor algebra TW of any Z 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 π 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 andup to isomorphism-to the Grabowski-Marmo brackets (the last bracket extends the Schouten-Nijenhuis 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.
Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of morphisms of Lie n-algebroids over different bases, give an explicit formula for the Chevalley-Eilenberg differential of a Lie n-algebroid, compare the categories of Lie n-algebroids and NQ-manifolds, and prove some conjectures of Sheng and Zhu [SZ11].where Perm denotes the set of all permutations (with repetitions) of the elements Addendum.It is straightforwardly checked that the signs (± 1 ) − (± 3 ) are given bywhich completes the explanation of Definition 6.
a b s t r a c tWe define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z 2 ) n -graded commutative associative algebra A. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (Z 2 ) n -graded matrices of degree 0 is polynomial in its entries. In the case of the algebra A = H of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (Z 2 ) n -graded version of Liouville's formula.
In Physics and in Mathematics Z n 2 -gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the 'scalar product' of the involved Z n 2 -degrees. The Z n 2 -Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. In this article we develop the foundations of the theory: we define Z n 2 -supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of Z • 2 -supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n-fold vector bundle has a canonical 'superization' to a Z n 2 -supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Z n 2 -context.
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