The Hopf Algebra of Fliess Operators and Its Dual Pre-lie Algebra |
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| Authors: | Loïc Foissy |
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| Institution: | 1. Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, , Université du Littoral C\tote d'Opale, Centre Universitaire de la Mi-Voix , Calais , France foissy@lmpa.univ-littoral.fr |
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| 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, finite-dimensional, connected grading. Dually, the vector space ? ? x 0, x 1 ? 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 ? ? x 0, x 1 ? a Com-Pre-Lie algebra. We give a presentation of this object as a pre-Lie algebra. |
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| Keywords: | Fliess operators Hopf algebras Pre-Lie algebras |
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