A generalized Cayley–Hamilton theorem

Abstract: Let be a solvable Lie subalgebra of the Lie algebra ( as a vector space). Let be polynomials in the commuting variables with coefficients in . For matrices , let and letIn this paper, we prove that, for , if one value of the matrix-valued function (the value depends on the product order of the variables) is nilpotent, then, (a) all values of are nilpotent; (b) all values of (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras. [Copyright &y& Elsevier]