Coassociativity Breaking and Oriented Graphs

With each coassociative coalgebra, we associate an oriented graph. The coproduct ∆, obeying the coassociativity equation (∆ ⊗ id)∆ = (id ⊗ ∆)∆ is then viewed as a physical propagator which can convey information. We notice that such a coproduct is non local. To recover locality we have to break the coassociativity of the coproduct and restore it, in some sense by introducing another coproduct ∆̃. The coassociativity equation becomes the coassociativity breaking equation (∆̃⊗id)∆ = (id⊗∆)∆̃. A coalgebra equipped with such coproducts, will be called a L-coalgebra. We prove then that any oriented graph whose paths may be equipped with a probability measure, can be seen as derived from such an algebraic formalism. The aim of this article is to study the consequences of such a physical viewpoint into the algebraic formalism, especially for a unital algebra equipped with its flower graph. We will show also the consequences of the notion of curvature of a 1-cochain introduced by Quillen in this framework and a common point with the associative product of this algebra, homomorphisms, Leibnitz derivatives and Ito derivatives. As the concept of Ito derivative appears naturally in this framework it will be studied throughout this article. In an attempt to adapt what was done in the case of cyclic cocycles to the Ito case, we construct a di-superalgebra (notions due to Quillen and Loday) from the curvature of an Ito derivative. We show also that (pre)-dialgebras are in one to one correspondence with dendriform algebras equipped with one associative product. We generalise our concept of L-coalgebras to L-coalgebras of degree n and show that M2(k) and the quaternions algebra H are (Markov) L-Hopf algebras of degree 2. We go further by introducing the notion of probabilistic algebraic product, notion which comes naturally from graph theory and coalgebra (matrix product in relation with the graph of Sl(2)q , wedge product in relation with the oriented triangle graph). 1 1991 Mathematics Subject Classification: 16A24, 60J15, 05C20