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We attach to any linear endomorphism f of any vector space V a structure ofprelie algebra on the shuffle algebra T(V); we describe its enveloping algebra,the dual Hopf algebra and the associated group of characters. For f=Id\_V, wefind the algebra of formal diffeomorphisms, seen as a subalgebra of theConnes-Kreimer Hopf algebra of rooted trees in the context of QFT; for otherwell-chosen f, we obtain the groups of Fliess operators in Control Theory. Analgebraic structures of these Com-Prelie Hopf algebras is carried out:gradations, group of automorphisms, subobject generated by V, etc.

A prelie algebra associated to a linear endomorphism and related algebraic structures