The Real Jacobi Group Revisited
The real Jacobi group GJ₁(ℝ), defined as the semi-direct product of the group SL(2, ℝ) with the Heisenberg group H₁, is embedded in a 4×4 matrix realisation of the group Sp(2, ℝ). The left-invariant one-forms on GJ₁(ℝ) and their dual orthogonal left-invariant vector fields are calculated in the S-co...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210292 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Real Jacobi Group Revisited / S. Berceanu // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 113 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862737781500936192 |
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| author | Berceanu, S. |
| author_facet | Berceanu, S. |
| citation_txt | The Real Jacobi Group Revisited / S. Berceanu // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 113 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The real Jacobi group GJ₁(ℝ), defined as the semi-direct product of the group SL(2, ℝ) with the Heisenberg group H₁, is embedded in a 4×4 matrix realisation of the group Sp(2, ℝ). The left-invariant one-forms on GJ₁(ℝ) and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates (x,y,θ,p,q,κ), and a left-invariant metric depending on 4 parameters (α,β,γ,δ) is obtained. An invariant metric depending on (α,β) in the variables (x,y,θ) on the Sasaki manifold SL(2, ℝ) is presented. The well-known Kähler balanced metric in the variables (x,y,p,q) of the four-dimensional Siegel-Jacobi upper half-plane XJ₁=GJ₁(ℝ)SO(2)×ℝ≈X₁×ℝ² depending on (α,γ) is written down as a sum of the squares of four invariant one-forms, where X₁ denotes the Siegel upper half-plane. The left-invariant metric in the variables (x,y,p,q,κ) depending on (α,γ,δ) of a five-dimensional manifold X~J₁=GJ₁(ℝ)SO(2)≈X₁×ℝ³ is determined.
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| first_indexed | 2025-12-07T21:25:02Z |
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| id | nasplib_isofts_kiev_ua-123456789-210292 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:02Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
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| spelling | Berceanu, S. 2025-12-05T09:22:52Z 2019 The Real Jacobi Group Revisited / S. Berceanu // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 113 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 32F45; 32Q15; 53C25; 53C22 arXiv: 1903.10721 https://nasplib.isofts.kiev.ua/handle/123456789/210292 https://doi.org/10.3842/SIGMA.2019.096 The real Jacobi group GJ₁(ℝ), defined as the semi-direct product of the group SL(2, ℝ) with the Heisenberg group H₁, is embedded in a 4×4 matrix realisation of the group Sp(2, ℝ). The left-invariant one-forms on GJ₁(ℝ) and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates (x,y,θ,p,q,κ), and a left-invariant metric depending on 4 parameters (α,β,γ,δ) is obtained. An invariant metric depending on (α,β) in the variables (x,y,θ) on the Sasaki manifold SL(2, ℝ) is presented. The well-known Kähler balanced metric in the variables (x,y,p,q) of the four-dimensional Siegel-Jacobi upper half-plane XJ₁=GJ₁(ℝ)SO(2)×ℝ≈X₁×ℝ² depending on (α,γ) is written down as a sum of the squares of four invariant one-forms, where X₁ denotes the Siegel upper half-plane. The left-invariant metric in the variables (x,y,p,q,κ) depending on (α,γ,δ) of a five-dimensional manifold X~J₁=GJ₁(ℝ)SO(2)≈X₁×ℝ³ is determined. This research was conducted in the framework of the ANCS project programs PN 16 42 01 01/2016, 18 09 01 01/2018, 19 06 01 01/2019. I had the idea to apply Lemma A.19 after the talk of Professor Zdaněk Dušek at the 1st International Conference on Differential Geometry (April 11-15, 2016, Fez, Morocco). I am grateful to Professor Zdanˇek for his correspondence in the first stages of the preparation of this paper. I would also like to thank Professor Mohamed Tahar Kadaoul Abbassi for the hospitality during the Fez conference and the partial financial support. I would like to thank Professor G.W. Gibbons for answering an email. I am grateful to Professor M. Visinescu for introducing me to the world of Sasaki manifolds. Thanks are also addressed to Professor R.D. Grigore for suggestions on some calculations. I am grateful to professors Dmitri Alekseevsky and Vicente Cortés for their criticism and suggestions on the first version of this paper. The author thanks the unknown referees who, through their recommendations, contributed to the improvement of the text of the paper. The author thanks Drs. I. Berceanu and M. Babalic for help in the preparation of the text. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Real Jacobi Group Revisited Article published earlier |
| spellingShingle | The Real Jacobi Group Revisited Berceanu, S. |
| title | The Real Jacobi Group Revisited |
| title_full | The Real Jacobi Group Revisited |
| title_fullStr | The Real Jacobi Group Revisited |
| title_full_unstemmed | The Real Jacobi Group Revisited |
| title_short | The Real Jacobi Group Revisited |
| title_sort | real jacobi group revisited |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210292 |
| work_keys_str_mv | AT berceanus therealjacobigrouprevisited AT berceanus realjacobigrouprevisited |