Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions
Abstract
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 string!
y black holes is also outlined.- Publication:
-
SIGMA
- Pub Date:
- June 2008
- DOI:
- 10.3842/SIGMA.2008.050
- arXiv:
- arXiv:0803.4436
- Bibcode:
- 2008SIGMA...4..050S
- Keywords:
-
- geometry over rings;
- non-commutative ring of order eight;
- Fano plane;
- Mathematical Physics;
- Mathematics - Algebraic Geometry;
- Quantum Physics
- E-Print:
- 6 pages, 1 table, 1 figure