Representations of Lie Algebras and Partial Differential Equations

TL;DR

This work explores explicit representations of simple Lie algebras, their connections to partial differential equations, orthogonal codes, combinatorics, and algebraic varieties, introducing new functions and functors in the field.

Contribution

It introduces new functors between Lie algebra representation categories, links PDEs to representation problems, and constructs orthogonal codes from Lie algebra representations.

Findings

Weight matrices generate large-distance orthogonal codes.

New hypergeometric functions related to Lie algebra root systems.

Irreducibility of representations linked to algebraic varieties.

Abstract

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.