Flag Partial Differential Equations and Representations of Lie Algebras

Details Flag Partial Differential Equations and Representations of Lie Algebras Flag Partial Differential Equations and Representations of Lie Algebras

2007-06-28

arxiv.org/abs/0706.4195v1

Mathematics

Representation Theory

30pages

Scientific paper

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 $G_2$.

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