Graded manifolds of type $$\Delta $$ Δ and n-fold vector bundles

Graded manifolds of type $$\Delta $$ Δ and n-fold vector bundles Lett Math Phys https://doi.org/10.1007/s11005-018-1105-9 Graded manifolds of type  and n-fold vector bundles Elizaveta Vishnyakova Received: 25 April 2017 / Revised: 2 May 2018 / Accepted: 27 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract Vector bundles and double vector bundles, or twofold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of supergeometry and Z-graded man- ifolds of degree ≤ 2. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of Z-graded manifolds of degree ≤ 2 and the category of (double) vector bundles with additional structures. In this paper we study the geometrization process in the case of Z -graded manifolds of type , where  is a certain weight system and r is the rank of . We establish an equivalence between a subcategory of the category of n-fold vector bundles and the category of graded manifolds of type . Keywords Graded manifold · Lie algebroid · Double vector bundle http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Letters in Mathematical Physics Springer Journals

Graded manifolds of type $$\Delta $$ Δ and n-fold vector bundles

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Physics; Theoretical, Mathematical and Computational Physics; Complex Systems; Geometry; Group Theory and Generalizations
ISSN
0377-9017
eISSN
1573-0530
D.O.I.
10.1007/s11005-018-1105-9
Publisher site
See Article on Publisher Site

Abstract

Lett Math Phys https://doi.org/10.1007/s11005-018-1105-9 Graded manifolds of type  and n-fold vector bundles Elizaveta Vishnyakova Received: 25 April 2017 / Revised: 2 May 2018 / Accepted: 27 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract Vector bundles and double vector bundles, or twofold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of supergeometry and Z-graded man- ifolds of degree ≤ 2. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of Z-graded manifolds of degree ≤ 2 and the category of (double) vector bundles with additional structures. In this paper we study the geometrization process in the case of Z -graded manifolds of type , where  is a certain weight system and r is the rank of . We establish an equivalence between a subcategory of the category of n-fold vector bundles and the category of graded manifolds of type . Keywords Graded manifold · Lie algebroid · Double vector bundle

Journal

Letters in Mathematical PhysicsSpringer Journals

Published: Jun 4, 2018

References

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