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

Metod Saniga (1) , Hans Havlicek (2) , Michel R. P. Planat (3) , Petr Pracna (4)

1 Astronomical Institute of the Slovak Academy of Sciences
2 TUW - Institut fuer Diskrete Mathematik und Geometrie
3 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)
4 JH-INST - J. Heyrovsky Institute of Physical Chemistry

Given 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.

Mots clés: Geometry over Rings; Non-Commutative Ring of Order Eight; Fano Plane

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]; Géométrie algébrique [math.AG]

Metod Saniga, Hans Havlicek, Michel R. P. Planat, Petr Pracna. Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions. SIGMA, 2008, 4, 050, 7 p. ⟨10.3842/SIGMA.2008.050⟩. ⟨hal-00268147v3⟩