Differential Calculus on -Graded Manifolds

G. Sardanashvily and W. Wachowski

Journal of Mathematics, 2017, vol. 2017, 1-19

Abstract:

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over -graded commutative rings and on -graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on -graded manifolds. We follow the notion of an -graded manifold as a local-ringed space whose body is a smooth manifold . A key point is that the graded derivation module of the structure ring of graded functions on an -graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body . Accordingly, the Chevalley–Eilenberg differential calculus on an -graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on -graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of -graded bundles.

Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:8271562

DOI: 10.1155/2017/8271562

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