Foundations of the Lie-Santilli Isotopic theory

• J. V. Kadeisvili Istituto per la Ricerca di Base Castello Principe Pignateii-Italia

Palabras clave: isotopies, Lie-Santilli isoalgebras, isogroups and isorepresenlatlons

Resumen

Lie's theory in lts current formulation is linear, local and canonical. As such, it is inapplicable to a growing number of nonlinear, nonlocal and noncanonical systems in various fields. In this paper we review and develop a generalization of Lie's theory proposed by R.M. Santilli in 1978 then at Harvard University and today called Lie-Santilli isotopic theory or lsotheory for short. The latter theory is based on the so-called isotopieswhich are nonlinear, nonlocal and noncanonical maps of any given linear, local and canonical theory capable of reconstructing linearity, locality and canonicity in certain generalized spaces and fields. The emerging Lie-Santilli isotheory is remarkable because it presever the abstract axioms of Lie's theory while being applicable to nonlinear, nonlocal and noncanonical systems. We review the foundations of the Lie-SantilIi isoalgebras and isogroups; introduce seemingly novel advances in their structure and interconnections; and show that he Lie-SantilIi isotheory provides the invariance of all infinitely possible, signature-preserving. nonlinear, nonlocal and noncanonical deformations of conventional Euclidean, Minkowskian or Riemannian invariants. We finally indicate a number of applications and identify rather intriguing open mathematical problems.