A General Study Of Nonsymmetric Cubic Algebraic Structures

In this thesis,we give a systematic study of non-symmetric binary cubic alge-braic and coalgebraic structures,including the cubic associative algebra,the cubic dendriform algebra,the cubic Rota-Baxter algebra,the cubic Hom-dendriform algebra and the cubic coalgebra.This thesis consists of six chapters.Chapter one introduces the background of the project,and then gives the motivations and outline of the paper.Chapter two focuses on cubic associative algebras.Firstly,we introduce the notion of a cubic associative algebra and then explore the relationship between cubic associative algebras and associative algebras,that is,an associative algebra is a cubic associative algebra.We also give the connection between cubic asso-ciative algebra and totally associative 3-algebra.Chapter two further constructs free cubic associative algebras and then show that a cubic associative algebra is not necessarily an associative algebra.Moreover,we construct free operated cubic associative algebras.Chapter three introduces the concepts of dendriform algebras and Rota-Baxter algebras.We then set up a precise connection from Rota-Baxter algebras to den-driform algebras.Chapter four extends dendriform algebras to cubic dendriform algebras.There are two approaches to take.Firstly,we introduce the concept and property of type A cubic dendriform algebras,together with the link from dendriform algebras to type A cubic dendriform algebras.We then introduce the definition of type B cubic dendriform algebras through the connection with Rota-Baxter algebras.Fi-nally,we discuss the relation between two kinds of cubic dendriform algebras,that is,they are equivalent.Chapter five explores cubic Hom-dendriform algebras.Chapter six considers cubic coalgebras.