An Introduction to the Lie–Santilli Theory

We review the elements of the nonlinear, nonlocal and noncanonical, axiom-preserving isotopies of Lie's theory introduced by the physicist R. M. Santilli back in 1978 while at the Department of Mathematics of Harvard University. We then study the structure theory of isotopic algebras and groups, and show that the emerging covering of Lie's theory can provide the symmetries for all possible deformations of a given metric directly in the frame of the experimenter (direct universality). We also show that the explicit form of the symmetry transformations can be readily computed from the knowledge of the original symmetry and of the new metric. The theory is illustrated with the isotopies of the rotational, Lorentz and Poincaré symmetries and an outline of their applications.