Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions
Given a finite associative ring with unity, R, any free (left) cyclic submodule (FCS) generated by a unimodular (n + 1)-tuple of elements of R represents a point of the n-dimensional projective space over R. Suppose that R also features FCSs generated by (n + 1)-tuples that are not unimodular: what...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2008 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2008
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149031 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions / M. Saniga, H. Havlicek, M. Planat, P. Pracna // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 25 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862628334737817600 |
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| author | Saniga, M. Havlicek, H. Planat, M. Pracna, P. |
| author_facet | Saniga, M. Havlicek, H. Planat, M. Pracna, P. |
| citation_txt | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions / M. Saniga, H. Havlicek, M. Planat, P. Pracna // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 25 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Given a finite associative ring with unity, R, any free (left) cyclic submodule (FCS) generated by a unimodular (n + 1)-tuple of elements of R represents a point of the n-dimensional projective space over R. Suppose that R also features FCSs generated by (n + 1)-tuples that are not unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for n = 2 when R is the (unique) non-commutative ring of order eight. The corresponding geometry is dubbed a ''Fano-Snowflake'' due to its diagrammatic appearance and the fact that it contains the Fano plane in its center. There exist, in fact, two such configurations – each being tied to either of the two maximal ideals of the ring – which have the Fano plane in common and can, therefore, be viewed as twins. Potential relevance of these noteworthy configurations to quantum information theory and stringy black holes is also outlined.
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| first_indexed | 2025-11-30T09:14:16Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149031 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T09:14:16Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Saniga, M. Havlicek, H. Planat, M. Pracna, P. 2019-02-19T13:10:07Z 2019-02-19T13:10:07Z 2008 Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions / M. Saniga, H. Havlicek, M. Planat, P. Pracna // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 25 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 51C05; 51Exx https://nasplib.isofts.kiev.ua/handle/123456789/149031 Given a finite associative ring with unity, R, any free (left) cyclic submodule (FCS) generated by a unimodular (n + 1)-tuple of elements of R represents a point of the n-dimensional projective space over R. Suppose that R also features FCSs generated by (n + 1)-tuples that are not unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for n = 2 when R is the (unique) non-commutative ring of order eight. The corresponding geometry is dubbed a ''Fano-Snowflake'' due to its diagrammatic appearance and the fact that it contains the Fano plane in its center. There exist, in fact, two such configurations – each being tied to either of the two maximal ideals of the ring – which have the Fano plane in common and can, therefore, be viewed as twins. Potential relevance of these noteworthy configurations to quantum information theory and stringy black holes is also outlined. The work was supported by the VEGA grant agency projects Nos. 6070 and 7012, the CNRSSAV Project No. 20246 “Projective and Related Geometries for Quantum Information” and by the Action Austria-Slovakia project No. 58s2 “Finite Geometries Behind Hilbert Spaces”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions Article published earlier |
| spellingShingle | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions Saniga, M. Havlicek, H. Planat, M. Pracna, P. |
| title | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions |
| title_full | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions |
| title_fullStr | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions |
| title_full_unstemmed | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions |
| title_short | Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions |
| title_sort | twin ''fano-snowflakes'' over the smallest ring of ternions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149031 |
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