Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions

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dc.creatorSaniga, Metod
dc.creatorHavlicek, Hans
dc.creatorPlanat, Michel
dc.creatorPracna, Petr
dc.date2008-03-31
dc.date2008-06-04
dc.date.accessioned2026-07-07T12:17:58Z
dc.date.available2026-07-07T12:17:58Z
dc.descriptionGiven a finite associative ring with unity, $R$, any free (left) cyclic submodule (FCS) generated by a $uni$modular ($n+1$)-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.
dc.description6 pages, 1 table, 1 figure; v2 -- standard representation of the ring of ternions given, 1 figure and 3 references added; v3 -- published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
dc.identifierhttps://arxiv.org/abs/0803.4436
dc.identifierhttp://arxiv.org/abs/0803.4436
dc.identifierSIGMA 4 (2008) 050, 7 pages
dc.identifierdoi:10.3842/SIGMA.2008.050
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/212256
dc.subjectMathematical Physics
dc.subjectAlgebraic Geometry
dc.subjectQuantum Physics
dc.titleTwin "Fano-Snowflakes" Over the Smallest Ring of Ternions
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