Differential Calculus on N-Graded Manifolds
The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over N-graded commutative rings and on N-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutati...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2017/8271562 |
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| author | G. Sardanashvily W. Wachowski |
| author_facet | G. Sardanashvily W. Wachowski |
| author_sort | G. Sardanashvily |
| collection | DOAJ |
| description | The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over N-graded commutative rings and on N-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 Z2-graded manifolds. We follow the notion of an N-graded manifold as a local-ringed space whose body is a smooth manifold Z. A key point is that the graded derivation module of the structure ring of graded functions on an N-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body Z. Accordingly, the Chevalley–Eilenberg differential calculus on an N-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on N-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of N-graded bundles. |
| format | Article |
| id | doaj-art-cd1f9714444442fb960692b4eaea9e4e |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-cd1f9714444442fb960692b4eaea9e4e2025-02-03T05:45:39ZengWileyJournal of Mathematics2314-46292314-47852017-01-01201710.1155/2017/82715628271562Differential Calculus on N-Graded ManifoldsG. Sardanashvily0W. Wachowski1Department of Theoretical Physics, Moscow State University, Moscow 119999, RussiaDepartment of Theoretical Physics, Moscow State University, Moscow 119999, RussiaThe differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over N-graded commutative rings and on N-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 Z2-graded manifolds. We follow the notion of an N-graded manifold as a local-ringed space whose body is a smooth manifold Z. A key point is that the graded derivation module of the structure ring of graded functions on an N-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body Z. Accordingly, the Chevalley–Eilenberg differential calculus on an N-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on N-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of N-graded bundles.http://dx.doi.org/10.1155/2017/8271562 |
| spellingShingle | G. Sardanashvily W. Wachowski Differential Calculus on N-Graded Manifolds Journal of Mathematics |
| title | Differential Calculus on N-Graded Manifolds |
| title_full | Differential Calculus on N-Graded Manifolds |
| title_fullStr | Differential Calculus on N-Graded Manifolds |
| title_full_unstemmed | Differential Calculus on N-Graded Manifolds |
| title_short | Differential Calculus on N-Graded Manifolds |
| title_sort | differential calculus on n graded manifolds |
| url | http://dx.doi.org/10.1155/2017/8271562 |
| work_keys_str_mv | AT gsardanashvily differentialcalculusonngradedmanifolds AT wwachowski differentialcalculusonngradedmanifolds |