This article introduces a method, which starting from simple and quite
general mathematical data, allows to construct linear algebras of operators
which are, each of them, endowed with a bialgebra structure (coproduct and
counity). Moreover under some explicit and natural conditions on theses
mathematical data we obtain linear algebras of operators with the following
property: each of them, is either a Hopf algebra, or its bialgebra structure
determines a more abstract Hopf algebra associated with it. Finally, we
describe a more general abstract condition for theses bialgebras to admit a
unique associated Hopf algebra. The presentation is adapted to the cases where
the algebras of linear operators are not finitely generated. This article is
restricted to the exposition of the method of construction and to the proofs of
existence and uniqueness of the structures associated with each algebra of
linear operators that are constructed.
Nevertheless, it may be noticed that the ideals of relations associated with
bialgebras that are obtained determine an algebraic domain which is of
theoretical interest .