The Hopf Algebra of Fliess Operators and Its Dual Pre-lie Algebra
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In this article, the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case was studied and it was shown that it admits a graded, finte-dimensional, connected gradation.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.read more
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A combinatorial Hopf algebra for nonlinear output feedback control systems
TL;DR: In this article, a combinatorial description of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory is provided.
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Algebraic structures of $F$-manifolds via pre-Lie algebras
TL;DR: In this article, the authors relate the algebraic structure on the tangent sheaf of an F-manifold (weak Frobenius manifold) defined by Hertling and Manin to the operad of pre-Lie algebras: for the filtration of the ideal generated by the Lie bracket.
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SISO Output Affine Feedback Transformation Group and Its Faa di Bruno Hopf Algebra
TL;DR: The general goal of this paper is to identify a transformation group that can be used to describe a class of feedback interconnections involving subsystems which are modeled solely in terms of Chen--Fliess functional expansions or Fliess operators and are independent of the existence of any state space models.
References
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Free Lie Algebras
TL;DR: In this article, it was shown that the Lie algebra of Lie polynomials is the free Lie algebra, and that its enveloping algebra is the associative algebra of noncommutative polynomorphisms.
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Pre-Lie algebras and the rooted trees operad
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On the Lie enveloping algebra of a pre-Lie algebra
Jean-Michel Oudom,Daniel Guin +1 more
TL;DR: In this article, an associative product on the symmetric module S(L) of any pre-Lie algebra L is constructed, which is then transformed into a Hopf algebra which is isomorphic to the enveloping algebra of LLie.