Curved Casimir Operators and the BGG Machinery
We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on v...
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| Дата: | 2007 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2007
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/147189 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Curved Casimir Operators and the BGG Machinery / A. Cap, V. Soucek // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ. |
Репозитарії
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irk-123456789-1471892019-02-14T01:24:59Z Curved Casimir Operators and the BGG Machinery Cap, A. Soucek, V. We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence. 2007 Article Curved Casimir Operators and the BGG Machinery / A. Cap, V. Soucek // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 22E46; 53A40; 53C15; 58J70 http://dspace.nbuv.gov.ua/handle/123456789/147189 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence. |
| format |
Article |
| author |
Cap, A. Soucek, V. |
| spellingShingle |
Cap, A. Soucek, V. Curved Casimir Operators and the BGG Machinery Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Cap, A. Soucek, V. |
| author_sort |
Cap, A. |
| title |
Curved Casimir Operators and the BGG Machinery |
| title_short |
Curved Casimir Operators and the BGG Machinery |
| title_full |
Curved Casimir Operators and the BGG Machinery |
| title_fullStr |
Curved Casimir Operators and the BGG Machinery |
| title_full_unstemmed |
Curved Casimir Operators and the BGG Machinery |
| title_sort |
curved casimir operators and the bgg machinery |
| publisher |
Інститут математики НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147189 |
| citation_txt |
Curved Casimir Operators and the BGG Machinery / A. Cap, V. Soucek // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT capa curvedcasimiroperatorsandthebggmachinery AT soucekv curvedcasimiroperatorsandthebggmachinery |
| first_indexed |
2023-05-20T17:26:56Z |
| last_indexed |
2023-05-20T17:26:56Z |
| _version_ |
1796153317672353792 |