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

Abstract

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.

arXiv comment31 pages, third version

References in corpus (3)

• Pre-Lie algebras and the rooted trees operad
• On the Lie envelopping algebra of a pre-Lie algebra
• A rigidity theorem for preLie algebras