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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2010 |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2010
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146533 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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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nasplib_isofts_kiev_ua-123456789-146533 |
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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 |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Hopf Maps, Lowest Landau Level, and Fuzzy Spheres |
| spellingShingle |
Hopf Maps, Lowest Landau Level, and Fuzzy Spheres Hasebe, K. |
| title_short |
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_sort |
hopf maps, lowest landau level, and fuzzy spheres |
| author |
Hasebe, K. |
| author_facet |
Hasebe, K. |
| publishDate |
2010 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146533 |
| 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. |
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