The Hopf algebra of Fliess operators and its dual pre-Lie algebra

Loïc Foissy (1, 2)

1 LMR - Laboratoire de Mathématiques de Reims
2 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finte-dimensional, connected gradation. Dually, the vector space IR is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibility which makes IR a Com-pre-Lie algebra. We give a presentation of this object as a Com-pre-Lie algebra and as a pre-Lie algebra.

Mots clés: Fliess operators; pre-Lie algebras; Hopf algebras

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy. The Hopf algebra of Fliess operators and its dual pre-Lie algebra. Communications in Algebra, 2015. ⟨hal-00808513v3⟩