Bialgebra Realization (3). Induced Module, Envelopping Bialgebra, Dual. The Example of the Integration of Classical Vectors Fields

In this article an interpretation and a proof of some classical \\theorems in analysis on the integration of analytic vectors fields are derived from the algebraic method of realization of bialgebras which are constructed with the data of a linear application from a coalgebra into the algebra of right (or left) invariant operators on an approximated coalgebra. The results are obtained from these general algebraic construction and theorems by introducing the more restrictive notion of induced module. Then the associated envelopping bialgebra is defined and naturally belongs to the dual of the tensor algebra over the approximated coalgebra .An interesting technical contribution is due to the "coproduct", or coproducts given by the approximated coalgebra, in the classical case of vectors fields at least.