Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 050, 7 pages      arXiv:0803.4436      https://doi.org/10.3842/SIGMA.2008.050

Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions

Metod Saniga a, Hans Havlicek b, Michel Planat c and Petr Pracna d
a) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
b) Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria
c) Institut FEMTO-ST/CNRS, MN2S, 32 Avenue de l'Observatoire, F-25044 Besançon Cedex, France
d) J. Heyrovský Institute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejskova 3, CZ-182 23 Prague 8, Czech Republic

Received May 02, 2008, in final form May 30, 2008; Published online June 04, 2008

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 stringy black holes is also outlined.

Key words: geometry over rings; non-commutative ring of order eight; Fano plane.

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References

  1. Törner G., Veldkamp F.D., Literature on geometry over rings, J. Geom. 42 (1991), 180-200.
  2. Planat M., Saniga M., Kibler M.R., Quantum entanglement and projective ring geometry, SIGMA 2 (2006), 066, 14 pages, quant-ph/0605239.
  3. Saniga M., Planat M., Pracna P., Havlicek H., The Veldkamp space of two-qubits, SIGMA 3 (2007), 075, 7 pages, arXiv:0704.0495.
  4. Havlicek H., Saniga M., Projective ring line of a specific qudit, J. Phys. A: Math. Theor. 40 (2007), F943-F952, arXiv:0708.4333.
  5. Planat M., Baboin A.-C., Qudits of composite dimension, mutually unbiased bases and projective ring geometry, J. Phys. A: Math. Theor. 40 (2007), F1005-F1012, arXiv:0709.2623.
  6. Havlicek H., Saniga M., Projective ring line of an arbitrary single qudit, J. Phys. A: Math. Theor. 41 (2008), 015302, 12 pages, arXiv:0710.0941.
  7. Saniga M., Planat M., Pracna P., Projective ring line encompassing two-qubits, Theor. and Math. Phys. 155 (2008), 463-473, quant-ph/0611063.
  8. Saniga M., A fine structure of finite projective ring lines, an invited talk given at the workshop on Prolegomena for Quantum Computing (November 21-22, 2007, Besançon, France), slides of the talk are available at http://hal.archives-ouvertes.fr/hal-00199008.
  9. Veldkamp F.D., Projective planes over rings of stable rang 2, Geom. Dedicata 11 (1981), 285-308.
  10. Veldkamp F.D., Geometry over rings, in Handbook of Incidence Geometry, Editor F. Buekenhout, Elsevier, Amsterdam, 1995, 1033-1084.
  11. Herzer A., Chain geometries, in Handbook of Incidence Geometry, Editor F. Buekenhout, Elsevier, Amsterdam, 1995, 781-842.
  12. Benz W., Zur Umkehrung von Matrizen im Bereich der Ternionen, Mitt. Math. Ges. Hamburg 10 (1979), 509-512.
  13. Lex W., Poneleit V., Weinert H.J., Über die Einzigkeit der Ternionenalgebra und linksalternative Algebren kleinen Ranges, Acta Math. Acad. Sci. Hungar. 35 (1980), 129-138.
  14. Nöbauer C., The book of the rings - part I (2000), pages 65 and 76, available online from http://www.algebra.uni-linz.ac.at/~noebsi/pub/rings.ps.
  15. Polster B., A geometrical picture book, Springer, New York, 1998, Chapter 5.
  16. Brown E., The many names of (7,3,1), Math. Mag. 75 (2002), 83-94.
  17. Baez J., The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 145-205, math.RA/0105155.
  18. Lévay P., Stringy black holes and the geometry of entanglement, Phys. Rev. D 74 (2006), 024030, 16 pages, hep-th/0603136.
  19. Lévay P., Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane, Phys. Rev. D 75 (2007), 024024, 19 pages, hep-th/0610314.
  20. Duff M.J., Ferrara S., E6 and the bipartite entanglement of three qutrits, Phys. Rev. D 76 (2007), 124023, 7 pages, arXiv:0704.0507.
  21. Duff M.J., Ferrara S., E7 and the tripartite entanglement of seven qubits, Phys. Rev. D 76 (2007), 025018, 7 pages, quant-ph/0609227.
  22. Conway J.H., Elkies N.D., Martin J.L., The Mathieu group M12 and its pseudogroup extension M13, Experiment. Math. 15 (2006), 223-236, math.GR/0508630.
  23. Conway J.H., Simons C.S., 26 implies the bimonster, J. Algebra 235 (2001), 805-814.
  24. Bañados M., Teitelboim C., Zanelli J., The black hole in three dimensional space time, Phys. Rev. Lett. 69 (1992), 1849-1851, hep-th/9204099.
  25. Witten E., Three-dimensional gravity revisited, arXiv:0706.3359.

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