  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.
