, A Hopf-differential ideal of O J(G/B) is a Hopf ideal E of O J(G/B) such that X tot E ? E for every rational vector field X on B
, A differential algebraic subgroup H ? J(G/B) is a Zariski closed subset consisting of the zero locus of some Hopf-differential ideal E of
, As differential-Hopf ideals are radical (Remark A.1) we have that the natural correspondence between differential-Hopf ideals of O J(G/B) and differential algebraic subgroups of J(G/B) is, in fact
, A differential algebraic subgroup H ? J(G/B) is a differential algebraic group of G(U) defined over C(B) in the sense of Kolchin
We set m to be the dimension of fibers and n the dimension of S, S be smooth irreducible algebraic affine varieties and ? : M ? S be a bundle ,
, Definition B.1 Let D M ? End C C[M ] be the ring of operators generated by ? C[M ] acting by multiplication ? The set of derivations Der
, One has two different C[M ]-algebra structures on D M : 1. Left C[M ]-algebra structure given by C[M ] ? C D M ? D M such that f ? P ? f P where (f P )(g) = f P (g)
P ? f )(g) = P (f g). This action will be ,
, Then the jet bundle of V over M is the vector bundle whose sheaf of linear functions is given by D M ? ?C
, M ) is the linear sub-bundle L ? J(V /M ) annihilated by a D M -submodule N ? D M ? ?C[M ] ?(V * ). i and this implies Y (p) ? ker v(p) for v(p) ? End
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