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Communicated by B. BrosowskiLie's theory in its current formulation is linear, local and canonical. As such, it is not applicable to a growing number of non-linear, non-local and non-canonical systems which have recently emerged in particle physics, superconductivity, astrophysics and other fields. In this paper, which is written by a physicist for mathematicians, we review and develop a generalization of Lie's theory proposed by the Italian-American physicist R. M. Santilli back in 1978 when at the Department of Mathematics of Harvard University and today called Lie-Santilli isotheory. The latter theory is based on the so-called isotopies which are non-linear, non-local and non-canonical 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 preserves the abstract axioms of Lie's theory while being applicable to non-linear, non-local and non-canonical systems. After reviewing the foundations of the Lie-Santilli isoalgebras and isogroups, and introducing seemingly novel advances in their interconnections, we show that the Lie-Santilli isotheory provides the invariance of all infinitely possible (well-behaved), non-linear, non-local and non-canonical deformations of conventional Euclidean, Minkowskian or Riemannian invariants. We also show that the non-linear, non-local and non-canonical symmetry transformations of deformed invariants are easily computable from the linear, local and canonical symmetry transforms of the original invariants and the given deformation. We then briefly indicate a number of applications of the isotheory in various fields. Numerous rather fundamental and intriguing, open mathematical and physical problems are indicated during the course of our analysis.

Lie's theory in its current formulation is linear, local and canonical. As such, it is not applicable to a growing number of non‐linear, non‐local and non‐canonical systems which have recently emerged in particle physics, superconductivity, astrophysics and other fields. In this paper, which is written by a physicist for mathematicians, we review and develop a generalization of Lie's theory proposed by the Italian–American physicist R. M. Santilli back in 1978 when at the Department of Mathematics of Harvard University and today called Lie–Santilli isotheory. The latter theory is based on the so‐called isotopies which are non‐linear, non‐local and non‐canonical 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 preserves the abstract axioms of Lie's theory while being applicable to non‐linear, non‐local and non‐canonical systems. After reviewing the foundations of the Lie–Santilli isoalgebras and isogroups, and introducing seemingly novel advances in their interconnections, we show that the Lie–Santilli isotheory provides the invariance of all infinitely possible (well‐behaved), non‐linear, non‐local and non‐canonical deformations of conventional Euclidean, Minkowskian or Riemannian invariants. We also show that the non‐linear, non‐local and non‐canonical symmetry transformations of deformed invariants are easily computable from the linear, local and canonical symmetry transforms of the original invariants and the given deformation. We then briefly indicate a number of applications of the isotheory in various fields. Numerous rather fundamental and intriguing, open mathematical and physical problems are indicated during the course of our analysis.

In this paper, we outline the mathematical and physical foundation of the 1991 hypothesis by R. M. Santilli of IsoRedShift (IRS), namely, a frequency shift of light toward the red characterized by the loss of energy by light to a cold medium without any relative motion between the source, the medium and the observer; we outline the corresponding foundations of Santilli's IsoBlueShift (IBS), namely, a frequency shift of light toward the blue characterized by the acquisition of energy by light from a hot medium without relative motion; we show the compatibility of Santilli's IRS and IBS with the axioms of special relativity under their proper mathematical formulations; we review the original experimental confirmations of IRS by Santilli in 2010 for a blue laser light in a tube containing air at pressure; we review the experimental confirmations of both IRS and IBS by G. West and G. Amato in 2011; we review the confirmatory measurements by Santilli, West and Amato done in 2012 on the West coast of Florida on the IRS origin of the redness of the Sun in the transition from the Zenith to the horizon; we present, for the first time, additional confirmatory measurements of IRS of Sunlight from the Zenith to Sunset done by the authors at the island of Kos, Greece, on September 20, 2012; we present, also for the first time, additional confirmatory measurements of IRS of Sunlight from Sunrise to the Zenith done in Cocoa Beach, East Coast of Florida, on October 20, 2012; we review the compatibility of Santilli's 1991 IRS hypothesis with Zwicky's 1929 hypothesis of Tired Light and identify their difference in the process originating the redshift; we recall the fit of cosmological redshift done by P. LaViolette in 1986 with Zwicky's Tired Light hypothesis; we confirm Santilli's 2010 argument according to which the IRS origin of the redness of the Sun at the horizon without relative motion is visual evidence on the expected absence of the expansion of the universe and of related conjectures; and we present the dismissal of various objections against Santilli's IRS and its interpretation of the cosmological redshift. In essence, we agree with Santilli that cosmologists should follow Galileo's teaching by establishing cosmological models via experiments on Earth prior to their application to the universe.

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