Compact Lie Groups, Generalised Euler Angles, and Applications
This is mainly a review of an intense 15-year long collaboration between the authors on explicit realisations of compact Lie groups and their applications. Starting with an elementary example, we will illustrate the main idea at the foundation of the generalisation of the Euler parametrisation of &l...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/8/10/492 |
| _version_ | 1827647796620034048 |
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| author | Sergio Luigi Cacciatori Antonio Scotti |
| author_facet | Sergio Luigi Cacciatori Antonio Scotti |
| author_sort | Sergio Luigi Cacciatori |
| collection | DOAJ |
| description | This is mainly a review of an intense 15-year long collaboration between the authors on explicit realisations of compact Lie groups and their applications. Starting with an elementary example, we will illustrate the main idea at the foundation of the generalisation of the Euler parametrisation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> to any compact Lie group. Based on this, we will provide a very detailed reconstruction of the possible Euler parametrisation associated with the so-called symmetric embedding. Then, we will recall how such constructions are related to the Dyson integrals, providing a geometrical interpretation of the latter, at least in certain cases. This includes a short review on the main properties of simple Lie groups, algebras, and their representations. Finally, we will conclude with some applications to nuclear physics and to measure theory in infinite dimensions and discuss some open questions. |
| first_indexed | 2024-03-09T19:24:37Z |
| format | Article |
| id | doaj.art-2133d0154d024c1da8e732be4b3dd3f5 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-09T19:24:37Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-2133d0154d024c1da8e732be4b3dd3f52023-11-24T03:01:09ZengMDPI AGUniverse2218-19972022-09-0181049210.3390/universe8100492Compact Lie Groups, Generalised Euler Angles, and ApplicationsSergio Luigi Cacciatori0Antonio Scotti1Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, ItalyDipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, ItalyThis is mainly a review of an intense 15-year long collaboration between the authors on explicit realisations of compact Lie groups and their applications. Starting with an elementary example, we will illustrate the main idea at the foundation of the generalisation of the Euler parametrisation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> to any compact Lie group. Based on this, we will provide a very detailed reconstruction of the possible Euler parametrisation associated with the so-called symmetric embedding. Then, we will recall how such constructions are related to the Dyson integrals, providing a geometrical interpretation of the latter, at least in certain cases. This includes a short review on the main properties of simple Lie groups, algebras, and their representations. Finally, we will conclude with some applications to nuclear physics and to measure theory in infinite dimensions and discuss some open questions.https://www.mdpi.com/2218-1997/8/10/492Lie groupsEuler anglesrepresentationsDyson integralssymmetric spaces |
| spellingShingle | Sergio Luigi Cacciatori Antonio Scotti Compact Lie Groups, Generalised Euler Angles, and Applications Universe Lie groups Euler angles representations Dyson integrals symmetric spaces |
| title | Compact Lie Groups, Generalised Euler Angles, and Applications |
| title_full | Compact Lie Groups, Generalised Euler Angles, and Applications |
| title_fullStr | Compact Lie Groups, Generalised Euler Angles, and Applications |
| title_full_unstemmed | Compact Lie Groups, Generalised Euler Angles, and Applications |
| title_short | Compact Lie Groups, Generalised Euler Angles, and Applications |
| title_sort | compact lie groups generalised euler angles and applications |
| topic | Lie groups Euler angles representations Dyson integrals symmetric spaces |
| url | https://www.mdpi.com/2218-1997/8/10/492 |
| work_keys_str_mv | AT sergioluigicacciatori compactliegroupsgeneralisedeuleranglesandapplications AT antonioscotti compactliegroupsgeneralisedeuleranglesandapplications |