Generalized Lie Algebraic Geometry in R3x SO(3) Configuration Space for SU(3) of Elementary Particles and for Wave-packing of Atomic Structure

Abstract

Abstract In this paper, we show, by extending Lie’s original 1871 thesis “on philosophical reflections upon the nature of Cartesian geometry” based on “transformation by which surfaces that touch each other are turned into similar surfaces … between the Plucker line geometry and a geometry whose elements are the space’s spheres” to include toroidal deformation of the sphere, how an algebraic geometric principle of duality between points, lines and planes of 3..dimesional space provides a sufficiently general framework for realizations of Lie-algebra and its Lie-isotopic and Lie-admissible generalizations in solid state configuration space R3×SO(3) compatible with translational periodicity of 3–dimensional space lattice. The generalization provide not only representation of SU(3) symmetry of extended (string-like) elementary particles with complementary duality of leptons and baryons, but also dual wave-packet representation of atomic structure and the periodic table, highlighting the significance of the fact that Mendeleev originally moulded his two-dimensional rendering of the periodic system on the dual Sanskrit grammar/phonetics.