On Relative Tractor Bundles
This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated with the geometry in question. In many situations of interest, it is known that this...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212784 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On Relative Tractor Bundles. Andreas Čap, Zhangwen Guo and Michał Andrzej Wasilewicz. SIGMA 20 (2024), 108, 19 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862554870577364992 |
|---|---|
| author | Čap, Andreas Guo, Zhangwen Wasilewicz, Michał Andrzej |
| author_facet | Čap, Andreas Guo, Zhangwen Wasilewicz, Michał Andrzej |
| citation_txt | On Relative Tractor Bundles. Andreas Čap, Zhangwen Guo and Michał Andrzej Wasilewicz. SIGMA 20 (2024), 108, 19 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated with the geometry in question. In many situations of interest, it is known that this sequence provides a resolution of a sheaf that can locally be realized as a pullback from a local leaf space of a foliation that is naturally available in this situation. An explicit description of the latter sheaf was only available under much more restrictive assumptions. For any geometry that admits relative tractor bundles, we construct a large family of such bundles for which we obtain a simple, explicit description of the resolved sheaves under weak assumptions on the torsion of the geometry. In particular, we discuss the cases of Legendrean contact structures and of generalized path geometries, which are among the most important examples for which the relative BGG machinery is available. In both cases, we show that essentially all relative tractor bundles are obtained by our construction, and our description of the resolved sheaves applies whenever the BGG sequence is a resolution.
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| first_indexed | 2026-03-21T11:37:40Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212784 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:37:40Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Čap, Andreas Guo, Zhangwen Wasilewicz, Michał Andrzej 2026-02-11T10:38:03Z 2024 On Relative Tractor Bundles. Andreas Čap, Zhangwen Guo and Michał Andrzej Wasilewicz. SIGMA 20 (2024), 108, 19 pages 1815-0659 2020 Mathematics Subject Classification: 58J10; 53C07; 53C15; 58J60; 58J70 arXiv:2405.13614 https://nasplib.isofts.kiev.ua/handle/123456789/212784 https://doi.org/10.3842/SIGMA.2024.108 This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated with the geometry in question. In many situations of interest, it is known that this sequence provides a resolution of a sheaf that can locally be realized as a pullback from a local leaf space of a foliation that is naturally available in this situation. An explicit description of the latter sheaf was only available under much more restrictive assumptions. For any geometry that admits relative tractor bundles, we construct a large family of such bundles for which we obtain a simple, explicit description of the resolved sheaves under weak assumptions on the torsion of the geometry. In particular, we discuss the cases of Legendrean contact structures and of generalized path geometries, which are among the most important examples for which the relative BGG machinery is available. In both cases, we show that essentially all relative tractor bundles are obtained by our construction, and our description of the resolved sheaves applies whenever the BGG sequence is a resolution. For open access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission. The first and second authors were supported by the Austrian Science Fund (FWF), doi:10.55776/P33559. The second and third authors were partially supported by the Austrian Science Fund (FWF), doi:10.55776/Y963. This article is based on work from COST Action CaLISTA CA21109 supported by COST (European Cooperation in Science and Technology, https://www.cost.eu). We thank the anonymous referees for several helpful comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Relative Tractor Bundles Article published earlier |
| spellingShingle | On Relative Tractor Bundles Čap, Andreas Guo, Zhangwen Wasilewicz, Michał Andrzej |
| title | On Relative Tractor Bundles |
| title_full | On Relative Tractor Bundles |
| title_fullStr | On Relative Tractor Bundles |
| title_full_unstemmed | On Relative Tractor Bundles |
| title_short | On Relative Tractor Bundles |
| title_sort | on relative tractor bundles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212784 |
| work_keys_str_mv | AT capandreas onrelativetractorbundles AT guozhangwen onrelativetractorbundles AT wasilewiczmichałandrzej onrelativetractorbundles |