Comments on the Regular and Irregular IsoRepresentations of the Lie-Santilli IsoAlgebras

Abstract

As it is well known, 20th century applied mathematics with related physical and chemical theories, are solely applicable to point-like particles moving in vacuum under Hamiltonian interactions (exterior dynamical problems). In this note, we study the covering of 20th century mathematics discovered by R. M. Santilli, today known as Santilli isomathematics, representing particles as being extended, non-spherical and deformable while moving within a physical medium under Hamiltonian and non-Hamiltonian interactions (interior dynamical problems). In particular, we focus the attention on a central part of isomathematics given by the isorepresentations of the Lie-Santilli isoalgebras that have been classified into regular (irregular) isorepresentations depending on whether the structure quantities of the isocommutation rules are constants (functions of local variables). The importance of the study of the isorepresentation theory for a number of physical and chemical applications is pointed out.

Exact and Invariant Representation of Nuclear Magnetic Moments and Spins According to Hadronic Mechanics

“…[22]; and Refs. [24][25][26][27][28][29][30][31][32][33] where quantities with a "hat" are formulated on isospaces over isofields and those without are their projection on conventional spaces over conventional fields.…”

“…The above basic assumptions imply the applicability of the Lie-Santilli isotheory [3b, 7,22,[24][25][26][27][28][29][30][31][32][33] at large that was constructed precisely for the representation of non-Hamiltonian systems under the most general known linear and non-linear, local and non-local and Hamiltonian as well as non-Hamiltonian interactions, Finally, the above basic assumptions imply that the universal symmetry for the non-relativistic treatment of isolated and stable nuclei is the Galileo-Santilli isosymmetry [21,22], while that for the relativistic treatment is the Lorentz-Poincaré-Santilli isosymmetry [12][13][14][15][16][17][18][19][20][21][22][23]. We can, therefore, introduce the following: DEFINITION A.1 [18,21,22]: A non-relativistic ˆ= 1 / > 0 I T…”

“…This occurrence is easily seen by specializing the isoline element (27) to the isolight isocone [23,37] (A. 48) thus leading to the Maximal Causal Speed of IsoAxiom 5.1 below.…”

“…In view of these advances, the isotopies of Lie's theory are today called the Lie-Santilli isotheory (see independent studies [24][25][26][27][28][29][30][31][32][33]). …”