Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formu...
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| Дата: | 2015 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2015
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/146839 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1468392019-02-12T01:24:01Z Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras Aizawa, N. Chandrashekar, R. Segar, J. The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if ℓ=1 and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when ℓ≠1. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules. 2015 Article Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B10; 58J70 DOI:10.3842/SIGMA.2015.002 http://dspace.nbuv.gov.ua/handle/123456789/146839 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 |
The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if ℓ=1 and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when ℓ≠1. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules. |
| format |
Article |
| author |
Aizawa, N. Chandrashekar, R. Segar, J. |
| spellingShingle |
Aizawa, N. Chandrashekar, R. Segar, J. Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Aizawa, N. Chandrashekar, R. Segar, J. |
| author_sort |
Aizawa, N. |
| title |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_short |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_full |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_fullStr |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_full_unstemmed |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras |
| title_sort |
lowest weight representations, singular vectors and invariant equations for a class of conformal galilei algebras |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146839 |
| citation_txt |
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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