Weighted Infinitesimal Bialgebras And Related Topics

The concept of weighted infinitesimal bialgebras is an algebraic meaning of the nonhomogenous associative Yang-Baxter equation,which plays an important role in mathematics and mathematical physics.In this thesis,we mainly study the weighted infinitesimal bialgebra.More precisely,we investigate the basic properties of a weighted infinitesimal bialgebra and show some examples of this algebra.We explore the relationships between weighted infinitesimal bialgebras,operated algebras and pre-Lie algebras.This thesis contains eight chapters.In Chapter 1,we introduce the backgrounds,motivations and the recent developments of weighted infinitesimal bialgebras.For the completeness,we also recall some relative definitions and facts for this paper.In Chapter 2,we first recall the concept of weighted infinitesimal(unitary)bialgebras,which generalizes simultaneously the one introduced by Joni and Rota and the one initiated by Loday and Ronco.We then show that some well-known algebras possess a weighted infinitesimal(unitary)bialgebra.Some basic properties of weighted infinitesimal(unitary)bialgebras are also investigated.In Chapter 3,we study two versions of infinitesimal Hopf algebras—the infinitesimal Hopf algebras in the sense of Aguiar and the infinitesimal unitary counitary Hopf algebra in the view of Loday and Ronco.In Chapter 4,we explore the relationship between solutions of weighted AYBEs and weighted infinitesimal unitary bialgebras.We give a bi.jection between solutions of the associative Yang-Baxter equation of weight ? and Rota-Baxter operators of weight ? on matrix algebras.Finally,We show that any weighted quasitriangular infinitesimal unitary bialgebra has a dendriform algebraic structure.In Chapter 5,we introduce the concept of weighted infinitesimal Hopf modules and show that any module carries a natural structure of weighted infinitesimal unitary Hopf module over a weighted quasitriangular infinitesimal unitary bialgebra.In Chapter 6,we decorate planar rooted forests in a new way,and prove that the space of decorated planar forests,together with a coproduct and a set of grafting operations,is the free ?-cocycle infinitesimal unitary bialgebra(resp.Hopf algebra)of weight zero on a set.A combinatorial description of the coproduct is also given.As applications,we obtain the initial object in the category of cocycle infinitesimal unitary bialgebras(resp.Hopf algebras)on undecorated planar rooted forests,which is the object studied in the(noncommutative)Connes-Kreimer Hopf algebra.In Chapter 7,we introduce the concept of symmetric 1-cocycle conditions which is derived from a dual of the Hochschild cohomology.We study the universal properties of the space of decorated planar rooted forests in the framework of operated algebras,leading to the notation of a weighted ?-cocycle infinitesimal unitary bialgebra.We also construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco.In Chapter 8,we derive two pre-Lie algebras from an arbitrary weighted infinitesimal bialgebra and a weighted commutative infinitesimal bialgebra,respectively.The second construction generalizes the Gel'fand-Dorfman Theorem on Novikov algebras.As an application,a pre-Lie algebraic structure and then a new Lie algebraic structure on an associative algebra are constructed.Finally,we construct two new pre-Lie algebras on decorated planar rooted forests.