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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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Інститут математики НАН України
2008
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/149031 |
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irk-123456789-1490312019-02-20T01:28:49Z Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions Saniga, M. Havlicek, H. Planat, M. Pracna, P. 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. 2008 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149031 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| language |
English |
| 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. |
| format |
Article |
| author |
Saniga, M. Havlicek, H. Planat, M. Pracna, P. |
| spellingShingle |
Saniga, M. Havlicek, H. Planat, M. Pracna, P. Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Saniga, M. Havlicek, H. Planat, M. Pracna, P. |
| author_sort |
Saniga, M. |
| title |
Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions |
| title_short |
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_sort |
twin ''fano-snowflakes'' over the smallest ring of ternions |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149031 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT sanigam twinfanosnowflakesoverthesmallestringofternions AT havlicekh twinfanosnowflakesoverthesmallestringofternions AT planatm twinfanosnowflakesoverthesmallestringofternions AT pracnap twinfanosnowflakesoverthesmallestringofternions |
| first_indexed |
2023-05-20T17:32:02Z |
| last_indexed |
2023-05-20T17:32:02Z |
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1796153501447880704 |