Comparison of the Modular and Non-modular Cases

Abstract

The famous theorem of Lie on solvable linear Lie algebras over an algebraically closed field of characteristic zero asserts that every such algebra has a common eigenvector or, equivalently, consists of triangular matrices when interpreted as matrices relative to a suitable basis. The common algebraic proofs of this theorem use an argument which infers the nilpotency of a matrix from the vanishing of the traces of certain polynomials in the matrix, especially its powers, as in [234, pp. 43 – 50, and 64, pp. 2–05, 2–06]. One may also deduce the result from the corresponding one (due to Kolchin [254]) for solvable connected linear algebraic groups over arbitrary algebraically closed fields, via the correspondence in characteristic zero between linear algebraic groups and their Lie algebras [71, 72]. The conclusion fails for modular fields, although some of the proofs referred to are still applicable when the degree of the matrices is less than the characteristic. Over a field F of prime characteristic p, a 2-dimensional solvable Lie algebra L of p by p matrices is spanned by

.