Isotopic, Genotopic and Hyperstructural Liftings of Lie's Theory and their Isoduals

Abstract

After reviewing the basic role of Lie's theory for the mathematics and physics of this century, we identify its limitations for the treatment of systems beyond the local-differential, Hamiltonian and canonical-unitary conditions of the original conception. We therefore outline three sequential generalized mathematics introduced by the author under the name of iso-, geno- and hyper-mathematics which are based on generalized, Hermitean, non-Hermitean and multi-valued units, respectively. The resulting iso-, geno- and hyper-Lie theories, for which the new mathematics were submitted, have been extensively used for the description of nonlocal-integral systems with action-at-a-distance Hamiltonian and short-range-contact non-Hamiltonian interactions in reversible, irreversible and multi-valued conditions, respectively. We then point out that conventional, iso-, geno- and hyper-Lie theories are unable to provide a consistent classical representation of antimatter yielding the correct charge conjugate states at the operator counterpart. We therefore outline yet novel mathematics proposed by the author under the names of isodual conventional, iso-, geno- and hyper-mathematics, which constitute anti-isomorphic images of the original mathematics characterized by negative-definite units and norms. The emerging isodual generalizations of Lie's theory have permitted a novel consistent characterization of antimatter at all levels of study, from Newton to second quantization. The main message emerging after three decades of investigations is that the sole generalized theories as invariant as the original formulation, the sole usable in physics, are those preserving the original abstract Lie axioms, and merely realizing them in broader forms.