Graded Bundles in the Category of Lie Groupoids
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrica...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147161 |
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| Cite this: | Graded Bundles in the Category of Lie Groupoids / A.J. Bruce, K. Grabowska, J. Grabowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 46 назв. — англ. |
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Bruce, A.J. Grabowska, K. Grabowski, J. 2019-02-13T17:58:32Z 2019-02-13T17:58:32Z 2015 Graded Bundles in the Category of Lie Groupoids / A.J. Bruce, K. Grabowska, J. Grabowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 46 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22A22; 55R10; 58E40; 58H05 DOI:10.3842/SIGMA.2015.090 https://nasplib.isofts.kiev.ua/handle/123456789/147161 We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that VB-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids, objects defined in a previous publication of the authors, which are graded manifolds in the category of Lie algebroids, showing that they are naturally related via differentiation and integration. In this work we also make an initial study of weighted Poisson-Lie groupoids and weighted Lie bi-algebroids, as well as weighted Courant algebroids. This paper is a contribution to the Special Issue on Poisson Geometry in Mathematics and Physics. The full collection is available at http://www.emis.de/journals/SIGMA/Poisson2014.html. The authors are indebted to the anonymous referees whose comments have served to improve the content and presentation of this paper. The research of K. Grabowska and J. Grabowski was funded by the Polish National Science Centre grant under the contract number DEC2012/06/A/ST1/00256. A.J. Bruce graciously acknowledges the financial support of the Warsaw Centre for Mathematics and Computer Science in the form of a postdoctoral fellowship. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Graded Bundles in the Category of Lie Groupoids Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Graded Bundles in the Category of Lie Groupoids |
| spellingShingle |
Graded Bundles in the Category of Lie Groupoids Bruce, A.J. Grabowska, K. Grabowski, J. |
| title_short |
Graded Bundles in the Category of Lie Groupoids |
| title_full |
Graded Bundles in the Category of Lie Groupoids |
| title_fullStr |
Graded Bundles in the Category of Lie Groupoids |
| title_full_unstemmed |
Graded Bundles in the Category of Lie Groupoids |
| title_sort |
graded bundles in the category of lie groupoids |
| author |
Bruce, A.J. Grabowska, K. Grabowski, J. |
| author_facet |
Bruce, A.J. Grabowska, K. Grabowski, J. |
| publishDate |
2015 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that VB-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids, objects defined in a previous publication of the authors, which are graded manifolds in the category of Lie algebroids, showing that they are naturally related via differentiation and integration. In this work we also make an initial study of weighted Poisson-Lie groupoids and weighted Lie bi-algebroids, as well as weighted Courant algebroids.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147161 |
| citation_txt |
Graded Bundles in the Category of Lie Groupoids / A.J. Bruce, K. Grabowska, J. Grabowski // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 46 назв. — англ. |
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AT bruceaj gradedbundlesinthecategoryofliegroupoids AT grabowskak gradedbundlesinthecategoryofliegroupoids AT grabowskij gradedbundlesinthecategoryofliegroupoids |
| first_indexed |
2025-12-01T19:55:42Z |
| last_indexed |
2025-12-01T19:55:42Z |
| _version_ |
1850860878818181120 |