Lie algebras associated with the exponential solutions of nonautonomous linear differential equations

A new free Lie algebra structure generated by a continuous set of operators is introduced. The Friedrichs' criterion, which is known for a finite (cou ntable) number of generators, is extended to the continuous case. Based on Friedrichs' criterion, we obtain an explicit expression for the coefficient s of the Magnus expansion, which is central for one to get approximately th e exponential solutions of nonautonomous linear differential equations. (C) 2001 American Institute of Physics.