Hopf Cobrace And Its Construction

Yang-Baxter equation is the basic equation of mathematics and physics,which also lays the foundation of quantum group theory.Quantum Yang-Baxter equation plays an important role in the development of Hopf algebra,so it is very basic and important to find solutions of the Yang-Baxter equation.In order to study the non-degenerate and involutive solution of Yang-Baxter equation,Rump introduced the concept of brace for the first time,which is a generalization of Jacobson radical rings.After the concept of brace was proposed,the concepts of skew brace and semi-brace were introduced successively.These structures were used to construct non-involutive solutions of Yang-Baxter equation.Through their connection with the Yang-Baxter equation and group theory,brace has aroused the attention and research of many scholars,and has been developed rapidly.Hopf brace is a natural generalization form of brace and semi-brace,and is a new algebraic structure related to Yang-Baxter equation.On this basis,this paper introduced and studied the concept of Hopf cobrace with another new coalgebra structure,and established the relationship between Hopf cobrace and Yang-Baxter equation.This thesis is divided into five chapters:In Chapter 1,we gave the research background of this thesis,introduced the development and theoretical basis of braces,and gave the main results of this paper.In Chapter 2,we introduced the examples and properties of Hopf cobraces and showed the equivalence between the full subcategory of the category of bijective 1-cocycle and Hopf cobrace category.In Chapter 3,we mainly studied the commutative Hopf cobraces and established the relationship between Hopf cobrace and solution of Yang-Baxter equation.In Chapter 4,we mainly showed the equivalence between Hopf cobrace category and Hopf matched pair category.In Chapter 5,we mainly constructed Hopf cobraces.