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Theses

Algebraic descriptions of weighted directed graphs and their applications

Abstract : One of the main goal of this thesis is the construction of an algebraic formalism embedding the notion of weighted directed graph and that of coassociative coalgebra. We show the necessity to work with coalgebras equipped with two coproducts or co-operations. Therefore, we recover the notion of associative dialgebra introduced ten years ago by Jean-Louis Loday, notion motivated by K-theory, proposing thus a complementary point of view to his formalism. The development of the algebraic formalism introduced in this thesis also proposes an extension of the notion of directed graphs and demonstrate the usefulness of coalgebras equipped with several coproducts. The constuction of directed graphs on algebraic objects such as algebras, bialgebras or Hopf algebras motivate the natural construction of other types of algebras such as associative trialgebras, cubical trialgebras, dendriform algebras, dipterous algebras and pre-dendriform algebras introduced by Jean-Louis Loday and Maria Ronco and re-discovered by the author. The construction of tilings or coassociative coverings of directed graphs by coassociative coalgebras or codipterous coalgebras allows the construction of more general algebraic objects named coassociative manifolds. Other objectives developed in this thesis are related to coassociative grammars or to the Quillen formalism applied to Leibniz-Ito derivatives, a tool arising in classical and quantum stochastic calculi and naturally related to the author's framework.
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https://tel.archives-ouvertes.fr/tel-00007375
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Submitted on : Wednesday, December 8, 2004 - 11:11:35 AM
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  • HAL Id : tel-00007375, version 1

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Philippe Leroux. Algebraic descriptions of weighted directed graphs and their applications. Mathematics [math]. Université Rennes 1, 2003. English. ⟨tel-00007375⟩

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