Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegra...
Збережено в:
| Дата: | 2015 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2015
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/147106 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems / R. Heinonen, E.G. Kalnins, W. Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 34 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-147106 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1471062019-02-15T01:24:50Z Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Inönü-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ℏ→0 and nonrelativistic phenomena from special relativistic as c→∞, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras e(2,C) in flat space and o(3,C) on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of Inönü-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. After constant curvature spaces, the 4 Darboux spaces are the 2D manifolds admitting the most 2nd order Killing tensors. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by Inönü-Wigner contractions. We present tables of the contraction results. 2015 Article Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems / R. Heinonen, E.G. Kalnins, W. Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E70; 16G99; 37J35; 37K10; 33C45; 17B60 DOI:10.3842/SIGMA.2015.043 http://dspace.nbuv.gov.ua/handle/123456789/147106 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 |
Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Inönü-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ℏ→0 and nonrelativistic phenomena from special relativistic as c→∞, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras e(2,C) in flat space and o(3,C) on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of Inönü-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. After constant curvature spaces, the 4 Darboux spaces are the 2D manifolds admitting the most 2nd order Killing tensors. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by Inönü-Wigner contractions. We present tables of the contraction results. |
| format |
Article |
| author |
Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. |
| spellingShingle |
Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Heinonen, R. Kalnins, E.G. Miller Jr., W. Subag, E. |
| author_sort |
Heinonen, R. |
| title |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_short |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_full |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_fullStr |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_full_unstemmed |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems |
| title_sort |
structure relations and darboux contractions for 2d 2nd order superintegrable systems |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147106 |
| citation_txt |
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems / R. Heinonen, E.G. Kalnins, W. Miller Jr., E. Subag // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 34 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT heinonenr structurerelationsanddarbouxcontractionsfor2d2ndordersuperintegrablesystems AT kalninseg structurerelationsanddarbouxcontractionsfor2d2ndordersuperintegrablesystems AT millerjrw structurerelationsanddarbouxcontractionsfor2d2ndordersuperintegrablesystems AT subage structurerelationsanddarbouxcontractionsfor2d2ndordersuperintegrablesystems |
| first_indexed |
2023-05-20T17:26:36Z |
| last_indexed |
2023-05-20T17:26:36Z |
| _version_ |
1796153296968220672 |