Structure fractals and para-quaternionic geometry

Julian Ławrynowicz; Massimo Vaccaro

Annales UMCS, Mathematica (2011)

  • Volume: 65, Issue: 2, page 63-73
  • ISSN: 2083-7402

Abstract

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It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p ← p + 2 ← p + 4 ← …, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens". The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.

How to cite

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Julian Ławrynowicz, and Massimo Vaccaro. "Structure fractals and para-quaternionic geometry." Annales UMCS, Mathematica 65.2 (2011): 63-73. <http://eudml.org/doc/268160>.

@article{JulianŁawrynowicz2011,
abstract = {It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p ← p + 2 ← p + 4 ← …, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens". The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.},
author = {Julian Ławrynowicz, Massimo Vaccaro},
journal = {Annales UMCS, Mathematica},
keywords = {Fractal; quaternion; para-quaternion; Clifford structure; para-quaternionic structure; bilinear form; quadratic form; fractal},
language = {eng},
number = {2},
pages = {63-73},
title = {Structure fractals and para-quaternionic geometry},
url = {http://eudml.org/doc/268160},
volume = {65},
year = {2011},
}

TY - JOUR
AU - Julian Ławrynowicz
AU - Massimo Vaccaro
TI - Structure fractals and para-quaternionic geometry
JO - Annales UMCS, Mathematica
PY - 2011
VL - 65
IS - 2
SP - 63
EP - 73
AB - It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p ← p + 2 ← p + 4 ← …, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens". The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.
LA - eng
KW - Fractal; quaternion; para-quaternion; Clifford structure; para-quaternionic structure; bilinear form; quadratic form; fractal
UR - http://eudml.org/doc/268160
ER -

References

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  1. Kigami, J., Analysis on Fractals, Cambridge Tracts in Mathematics 143, Cambridge University Press, Cambridge, 2001. Zbl0998.28004
  2. Kovacheva, R., Ławrynowicz, J. and Nowak-Kępczyk, M., Critical dimension 13 in approximation related to fractals of algebraic structure, XIII Conference on Analytic Functions, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 37 (2002), 77-102. Zbl1088.30512
  3. Ławrynowicz, J., Marchiafava, S. and Nowak-Kępczyk, M., Quaternionic background of the periodicity of petal and sepal structures in some fractals of the flower type, Proceedings of the 5th ISAAC Congress, Catania, July 25-30, 2005: More Progresses in Analysis, ed. by H. Begehr and F. Nicolosi, World Scientific, Singapore, 2009, pp. 987-996. Zbl1181.81072
  4. Ławrynowicz, J., Marchiafava, S. and Nowak-Kępczyk, M., Periodicity theorem for structure fractals in quaternionic formulation, Int. J. Geom. Methods Mod. Phys. 5 (2006), 1167-1197. Zbl1114.81047
  5. Ławrynowicz, J., Marchiafava, S. and Nowak-Kępczyk, M., Cluster sets and periodicity in some structure fractals, Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, ed. by S. Dimiev and K. Sekigawa, World Scientific, New Jersey-Singapore-London, 2007, pp. 179-195. Zbl1135.28006
  6. Ławrynowicz, J., Marchiafava, S. and Nowak-Kępczyk, M., Mathematical outlook of fractals and related to simple orthorhombic Ising-Onsager-Zhan lattices, Complex Strucures, Intergrability, Vector Fields, and Mathematical Physics, ed. by S. Dimiev and K. Sekigawa, World Scientific, Singapore, 2009, pp. 156-166. Zbl1186.82018
  7. Ławrynowicz, J., Marchiafava, S. and Nowak-Kępczyk, M., Applied peridicity theorem for structure fractals in quaternionic formulation, Applied Complex and Quaternionic Approximation, ed. by R. K, Kovacheva, J. Ławrynowicz and S. Marchiafava, Ediz. Nuova Cultura Univ. "La Sapienza", Roma, 2009, pp. 93-122. 
  8. Ławrynowicz, J., Nôno, K. and Suzuki, O., Hyperfunctions on fractal boundaries: meromorphic Schauder basis for a fractal set, Bull. Soc. Sci. Lett. Łódź Sér. Déform. 53 (2007), 63-74. Zbl1152.46027
  9. Ławrynowicz, J., Nowak-Kępczyk, M. and Suzuki, O., A duality theorem for inoculated graded fractal bundles vs. Cuntz algebras and their central extensions, Int. J. Pure Appl. Math. 52 (2009), 315-338. Zbl1171.81394
  10. Ławrynowicz, J., Ogata, T. and Suzuki, O., Differential and integral calculus for Schauder basis on a fractal set (I) (Schauder basis 80 years after), Lvov Mathematical School in the Period 1915-45 as Seen Today, ed. by B. Bojarski, J. Ławrynowicz and Ya. G. Prytuła, Banach Center Publications 87, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 2009, pp. 115-140. Zbl1218.37062
  11. Ławrynowicz, J., Suzuki, O., Periodicity theorem for graded fractal bundles related to Clifford structures, Int. J. Pure Appl. Math. 24 (2005), no. 2, 181-209. Zbl1160.28303
  12. Ławrynowicz, J., Suzuki, O. and Castillo Alvarado, F. L., Basic properties and applications of graded fractal bundles related to Clifford structures: An introduction, Ukrain. Math. Zh. 60 (2008), 603-618. Zbl1156.28305
  13. Lounesto, P., Clifford Algebra and Spinors, London Math. Soc. Lecture Notes 239, Cambridge Univ. Press, Cambridge, 1997, 2nd ed. (vol. 286), ibid. 2001. 
  14. Vaccaro, M., Subspaces of a para-quaternionic Hermitian vector space, 2010, ArXiv:1011.2947v1 [math. D6]; Internat. J. of Geom. Methods in Modern Phys. 8 (2011), in print. 
  15. Vaccaro, M., Basics of linear para-quaternionic geometry I. Hermitian para-type structure on a real vector space, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 61 (2011), no. 1, 23-36. Zbl1293.53039
  16. Vaccaro, M., Basics of linear para-quaternionic geometry II. Decomposition of a generic subspace of a para-quaternionic hermitian vector space, Bull. Soc. Sci. Lett. Łódź Sér. Rech. Déform. 61 (2011), no. 2, 17-34. Zbl1275.53033

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