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Structure fractals and para-quaternionic geometry. (English) Zbl 1247.81191

Summary: 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 \(n = 2^{7}\). Instead, J. Ławrynowicz and O. Suzuki [Int. J. Pure Appl. Math. 24, No. 2, 181–209 (2005; Zbl 1160.28303)] have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process \(p \to p + 2 \to p + 4 \to \ldots \), they have constructed \(2^{4}\)-dimensional “bipetals” for \(p = 9\) and \(2^{7}\)-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 author, M. Nowak-Kȩpczyk and S. Marchiafava Int. J. Geom. Methods Mod. Phys. 3, No. 5–6, 1167–1197 (2006; Zbl 1114.81047); Sekigawa, Kouei (ed.) et al., Trends in differential geometry, complex analysis and mathematical physics. Proceedings of 9th international workshop on complex structures, integrability and vector fields. Hackensack, NJ: World Scientific. 156–166 (2009; Zbl 1186.82018); R. Kovacheva (ed.) et al., Applied peridicity theorem for structure fractals in quaternionic formulation, Appl. Complex and Quaternionic Approximation. Ediz. Nuova Cultura Univ. “La Sapienza”, Roma, 93–122 (2009)] 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.

MSC:

81R25 Spinor and twistor methods applied to problems in quantum theory
32L25 Twistor theory, double fibrations (complex-analytic aspects)
15A66 Clifford algebras, spinors
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
28A80 Fractals
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