  This book is mainly an exposition of the author's works and his joint works
with his former students on explicit representations of finite-dimensional
simple Lie algebras, related partial differential equations, linear orthogonal
algebraic codes, combinatorics and algebraic varieties. Various oscillator
generalizations of the classical representation theorem on harmonic polynomials
are presented. New functors from the representation category of a simple Lie
algebra to that of another simple Lie algebra are given. Partial differential
equations play key roles in solving certain representation problems. The weight
matrices of the minimal and adjoint representations over the simple Lie
algebras of types E and F are proved to generate ternary orthogonal linear
codes with large minimal distances. New multi-variable hypergeometric functions
related to the root systems of simple Lie algebras are introduced in connection
with quantum many-body system in one-dimension. Certain equivalent
combinatorial properties on representation formulas are found. Irreducibility
of representations are proved directly related to algebraic varieties.
