Flag Partial Differential Equations and Representations of Lie Algebras

TL;DR

This paper develops algebraic methods to solve various variable-coefficient partial differential equations, finds polynomial solutions, and explores their applications in representing Lie algebras.

Contribution

It introduces algebraic techniques for solving complex PDEs and constructs polynomial solutions, linking these solutions to Lie algebra representations.

Findings

Solved initial value problems for generalized wave equations.

Found polynomial solutions for specific PDEs including Klein-Gordon and Laplace equations.

Constructed polynomial representations of Lie algebras like sl(n), so(n), and G_2.

Abstract

In this paper, we solve the initial value problems of variable-coefficient generalized wave equations associated with trees and a large family of linear constant-coefficient partial differential equation by algebraic methods. Moreover, we find all the polynomial solutions for a 3-dimensional variable-coefficient flag partial differential equation of any order, the linear wave equation with dissipation and the generalized anisymmetrical Laplace equation. Furthermore, the polynomial-trigonometric solutions of a generalized Klein-Gordan equation associated with 3-dimensional generalized Tricomi operator $\ptl_x^2+x\ptl_y^2+y\ptl_z^2$ are also given. As applications to representations of Lie algebras, we find certain irreducible polynomial representations of the Lie algebras $sl(n,\mbb{F}), so(n,\mbb{F})$ and the simple Lie algebra of type $G_2$.