A pre-Lie algebra associated to a linear endomorphism and related algebraic structures

Abstract

We construct a functor $$T$$ from the category of endomorphisms of vector spaces to the category of Com-PreLie algebras. For any endomorphism $$f$$ of a vector space $$V$$ , we describe the enveloping algebra of the pre-Lie algebra $$T(V,f)$$ , the dual Hopf algebra and the associated group of characters. For $$f=\mathrm{Id}_V$$ , we find the algebra of formal diffeomorphisms, seen as a subalgebra of the Connes–Kreimer Hopf algebra of rooted trees in the context of QFT; for other well-chosen nilpotent $$f$$ , we obtain the groups of Fliess operators in Control Theory. An algebraic study of these Com-PreLie Hopf algebras is carried out: gradations, generation, subobject generated by $$V$$ , etc.