Flag Partial Differential Equations and Representations of Lie Algebras

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 $G_2$.

💡 Research Summary

The paper tackles a broad class of partial differential equations (PDEs) by exploiting algebraic structures rather than traditional analytic techniques. Its first major contribution is the solution of initial‑value problems for variable‑coefficient generalized wave equations that are encoded by rooted trees. By assigning a variable (x_i) to each vertex and a differential order (m_i) to each edge, the authors construct a non‑commutative differential operator
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