On Eigenvectors of Nilpotent Lie Algebras of Linear Operators

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector. Introduction Throughout this paper V is a vector space of positive dimension over a field f and g is a nilpotent Lie algebra over f of linear operators on V . An element u ∈ V is an eigenvector for S ⊂ g if u is an eigenvector for every operator in S. If V has a basis (e1, . . . , en) representing each element of g by an upper triangular matrix, then e1 is an eigenvector for g. Such a basis exists when f is algebraically closed and g is solvable (Lie’s Theorem), and also when every element of g is a nilpotent operator (Engel’s Theorem). Our results are further conditions guaranteeing existence eigenvectors. The minimal and characteristic polynomials of a linear operator A on V are denoted respectively by πA, μA ∈ f[t] = the ring of polynomials over f. written #S. Let k be a Galois extension field of f of degree d := [k : f], and define M ⊂ N to be the additive monoid generated by zero and the prime divisors d. Consider the conditions: (C1) μA splits in k for every A ∈ g (C2) dimV / ∈ M Our main result is: Theorem 1 If (C1) and (C2) hold then g has an eigenvector.