On the solution of linear differential equations in Lie groups

The subject matter of this paper is the solution of the linear differential equation y = a#t#y, y#0# = y0 , where y0 2 G, a# # #:R !gand g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus [16], we represent the solution as an infinite series whose terms are indexed by binary trees. This relationship between the infinite series and binary trees leads both to a convergence proof and to a constructive computational algorithm. This numerical method requires the evaluation of a large number of multivariate integrals but this can be accomplished in a tractable manner by using quadrature schemes in a novel manner and by exploiting the structure of the Lie algebra.