Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 247groups and Lie algebras have finite dimensional representations which arenot semisimple. An element of a semisimple Lie group or Lie algebra isitself semisimple if its image in every finite–dimensional representation issemisimple in the sense of matrices.Every semisimple Lie algebra g can be classified by its Dynkin diagram[Helgason (2001)].3.9 Lie Symmetries and Prolongations on ManifoldsIn this section we continue our expose on Lie groups of symmetry, as a linkto modern jet machinery, developed below.3.9.1 Lie Symmetry Groups3.9.1.1 Exponentiation of Vector Fields on MLet x = (x 1 , ..., x r ) be local coordinates at a point m on a smoothn−manifold M. Recall that the flow generated by the vector–fieldis a solution of the system of ODEsv = ξ i (x) ∂ x i ∈ M,dx idε = ξi (x 1 , ..., x m ),(i = 1, ..., r).The computation of the flow, or one–parameter group of diffeomorphisms,generated by a given vector–field v (i.e., solving the system of ODEs) isoften referred to as exponentiation of a vector–field, denoted by exp(εv) x(see [Olver (1986)]).If v, w ∈ M are two vectors defined bythenv = ξ i (x) ∂ x i and w = η i (x) ∂ x i,exp(εv) exp(θw) x = exp(θw) exp(εv) x,for all ɛ, θ ∈ R,x ∈ M, such that both sides are defined, iff they commute,i.e., [v, w] = 0 everywhere [Olver (1986)].A system of vector–fields {v 1 , ..., v r } on a smooth manifold M is ininvolution if there exist smooth real–valued functions h k ij (x), x ∈ M,