The Hopf algebra of Fliess operators and its dual pre-Lie algebra
Lo\"ic Foissy (LM-Reims, LMPA)

TL;DR
This paper explores the algebraic structures underlying Fliess operators in control theory, revealing a Hopf algebra and dual pre-Lie algebra with compatible structures, advancing the mathematical framework of control systems.
Contribution
It introduces a graded, finite-dimensional Hopf algebra of Fliess operators and characterizes its dual as a Com-pre-Lie algebra, providing new algebraic insights.
Findings
The Hopf algebra of Fliess operators is graded and finite-dimensional.
The dual space forms a Com-pre-Lie algebra with compatible structures.
The dual algebra admits a presentation as a Com-pre-Lie and 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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
