Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fu...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2010 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2010
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146533 |
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| Zitieren: | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres / K. Hasebe // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 102 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862714495718129664 |
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| author | Hasebe, K. |
| author_facet | Hasebe, K. |
| citation_txt | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres / K. Hasebe // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 102 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of ''compounds'' of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.
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| first_indexed | 2025-12-07T17:51:32Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146533 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:51:32Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hasebe, K. 2019-02-09T20:29:48Z 2019-02-09T20:29:48Z 2010 Hopf Maps, Lowest Landau Level, and Fuzzy Spheres / K. Hasebe // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 102 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B70; 58B34; 81V70 DOI:10.3842/SIGMA.2010.071 https://nasplib.isofts.kiev.ua/handle/123456789/146533 This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of ''compounds'' of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model. This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The ful`l collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html.
 I would like to thank Yusuke Kimura for collaborations. Many crucial ingredients in this review are based on the works with him. I am also indebted to Takehiro Azuma for email correspondence about mathematics of fuzzy spheres. Since this article is a review-type, many important works not performed by the author are included. Hereby, I express my gratitude to the researchers whose works are reviewed in the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hopf Maps, Lowest Landau Level, and Fuzzy Spheres Article published earlier |
| spellingShingle | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres Hasebe, K. |
| title | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres |
| title_full | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres |
| title_fullStr | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres |
| title_full_unstemmed | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres |
| title_short | Hopf Maps, Lowest Landau Level, and Fuzzy Spheres |
| title_sort | hopf maps, lowest landau level, and fuzzy spheres |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146533 |
| work_keys_str_mv | AT hasebek hopfmapslowestlandaulevelandfuzzyspheres |