arXiv subjects
Publications and source records attributed to Philippe Leroux.
We propose a generalisation of a recent work of M. Aguiar and J.-L. Loday on Quadri-algebras, called Ennea-algebras. In this second version, this paper has been extended. We show that the augmented free Ennea-algebra is a connected Hopf algebra and construct explicit formal deformations of dendriform dialgebras, quari-algebras and ennea-algebras via Baxter operators.
We construct via usual graph theory a class of associative dialgebras as well as a class of coassociative L-coalgebras. Tiling of the (n^2,1)-De bruijn graphs are also obtained and constructions of cubical trialgebras, (notion defined by J-L. Loday and M. Ronco) are given. We also obtain splitting of associative products into several ones.
In quantum information theory, for $a,b$ two positive operators living in $B(\mathcal{H})$, where $\mathcal{H}$ is a separable Hilbert space, the quantum fidelity is denoted by $a*b =(b^{1/2}ab^{1/2})^{1/2}$. One of the aim of this let ter is to interpret the quantum fidelity as an algebraic law. We remark that if $a,b,c$ are three positive operators whi ch commute pairwise, the law * becomes self-distributive, i.e. the third Reidemeister movement in knot theory is verif ied. We study the converse. Let three positive operators be given, does the fact that the third Reidemeister movement between them is possible implie that they commute pairwise ? Though in general we only conjecture it for the moment, we prove it in some par ticular but important cases. Should this movement be not possible, we interpret it as an obstruction to comm utativity. We give also new examples of quandle algebras and left distributive systems and study the generalisation of Ito maps.
This paper is an attempt to unify coassociative coalgebra theory and random walks on oriented graphs.