On an Isospectral Lie–Poisson System and Its Lie Algebra

Abstract

In this paper we analyze the matrix differential system X' = [N,X 2 ], where N is skew-symmetric and X(0) is symmetric. We prove that it is isospectral and that it is endowed with a Poisson structure, and we discuss its invariants and Casimirs. Formulation of the Poisson problem in a Lie-Poisson setting, as a flow on a dual of a Lie algebra, requires a computation of its faithful representation. Although the existence of a faithful representation, assured by the Ado theorem and a symbolic algorithm, due to de Graaf, exists for the general computation of faithful representations of Lie algebras, the practical problem of forming a "tight" representation, convenient for subsequent analytic and numerical work, belongs to numerical algebra. We solve it for the Poisson structure corresponding to the equation X' = [N,X 2 ].