Discrete Minimal Surface Algebras
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining r...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2010 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2010
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146344 |
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| Cite this: | Discrete Minimal Surface Algebras / J. Arnlind, J. Hoppe // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 17 назв. — англ. |
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Arnlind, J. Hoppe, J. 2019-02-09T09:00:43Z 2019-02-09T09:00:43Z 2010 Discrete Minimal Surface Algebras / J. Arnlind, J. Hoppe // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R10; 06B15 DOI:10.3842/SIGMA.2010.042 https://nasplib.isofts.kiev.ua/handle/123456789/146344 We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras. This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html. We would like to thank the Marie Curie Research Training Network ENIGMA and the Swedish Research Council, as well as the IHES, the Sonderforschungsbereich “Raum-Zeit-Materie” (SFB647) and ETH Z¨urich, for financial support respectively hospitality – and Martin Bordemann for many discussions and collaboration on related topics. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Discrete Minimal Surface Algebras Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Discrete Minimal Surface Algebras |
| spellingShingle |
Discrete Minimal Surface Algebras Arnlind, J. Hoppe, J. |
| title_short |
Discrete Minimal Surface Algebras |
| title_full |
Discrete Minimal Surface Algebras |
| title_fullStr |
Discrete Minimal Surface Algebras |
| title_full_unstemmed |
Discrete Minimal Surface Algebras |
| title_sort |
discrete minimal surface algebras |
| author |
Arnlind, J. Hoppe, J. |
| author_facet |
Arnlind, J. Hoppe, J. |
| publishDate |
2010 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Article |
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We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146344 |
| citation_txt |
Discrete Minimal Surface Algebras / J. Arnlind, J. Hoppe // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 17 назв. — англ. |
| work_keys_str_mv |
AT arnlindj discreteminimalsurfacealgebras AT hoppej discreteminimalsurfacealgebras |
| first_indexed |
2025-12-07T17:23:14Z |
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2025-12-07T17:23:14Z |
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1850871062077636608 |