Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration
We discuss a fine-tuning of the co- and contra-variant transforms through the construction of specific fiducial and reconstruction vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg group. The covariant transform provides intertwining operator...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211723 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration. Amerah A. Al Ameer and Vladimir V. Kisil. SIGMA 18 (2022), 065, 21 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862711782151290880 |
|---|---|
| author | Al Ameer, Amerah A. Kisil, Vladimir V. |
| author_facet | Al Ameer, Amerah A. Kisil, Vladimir V. |
| citation_txt | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration. Amerah A. Al Ameer and Vladimir V. Kisil. SIGMA 18 (2022), 065, 21 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We discuss a fine-tuning of the co- and contra-variant transforms through the construction of specific fiducial and reconstruction vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg group. The covariant transform provides intertwining operators between pairs of representations. In particular, we obtain the Zak transform as an induced covariant transform intertwining the Schrödinger representation on ₂(ℝ) and the lattice (nilmanifold) representation on ₂(²). Induced covariant transforms in other pairs are Fock-Segal-Bargmann and theta transforms. Furthermore, we describe peelings which map the group-theoretical induced representations to convenient representation spaces of analytic functions. Finally, we provide a condition that can be imposed on the reconstructing vector to obtain an intertwining operator from the induced contravariant transform.
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| first_indexed | 2026-03-19T16:05:37Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211723 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T16:05:37Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Al Ameer, Amerah A. Kisil, Vladimir V. 2026-01-09T12:46:36Z 2022 Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration. Amerah A. Al Ameer and Vladimir V. Kisil. SIGMA 18 (2022), 065, 21 pages 1815-0659 2020 Mathematics Subject Classification: 43A85; 47G10; 81R30 arXiv:2105.13811 https://nasplib.isofts.kiev.ua/handle/123456789/211723 https://doi.org/10.3842/SIGMA.2022.065 We discuss a fine-tuning of the co- and contra-variant transforms through the construction of specific fiducial and reconstruction vectors. The technique is illustrated on three different forms of induced representations of the Heisenberg group. The covariant transform provides intertwining operators between pairs of representations. In particular, we obtain the Zak transform as an induced covariant transform intertwining the Schrödinger representation on ₂(ℝ) and the lattice (nilmanifold) representation on ₂(²). Induced covariant transforms in other pairs are Fock-Segal-Bargmann and theta transforms. Furthermore, we describe peelings which map the group-theoretical induced representations to convenient representation spaces of analytic functions. Finally, we provide a condition that can be imposed on the reconstructing vector to obtain an intertwining operator from the induced contravariant transform. We are grateful to anonymous referees for many useful comments and remarks. Their suggestions were used for the paper’s improvements. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration Article published earlier |
| spellingShingle | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration Al Ameer, Amerah A. Kisil, Vladimir V. |
| title | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration |
| title_full | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration |
| title_fullStr | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration |
| title_full_unstemmed | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration |
| title_short | Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration |
| title_sort | tuning co- and contra-variant transforms: the heisenberg group illustration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211723 |
| work_keys_str_mv | AT alameerameraha tuningcoandcontravarianttransformstheheisenberggroupillustration AT kisilvladimirv tuningcoandcontravarianttransformstheheisenberggroupillustration |