The Group Of Brace And The Structure Of Hopf Cotruss

Quantum Yang-Baxter equation is one of the important equations in the field of mathematical physics.It has been concerned by many experts and scholars for many years and has become a hot topic of research.In order to explore the set solution of non-degenerate involution of Yang-Baxter equation,Rump first introduced the concept of brace.Since the concept of brace was proposed,brace has at.tracted the attention and research of many-experts and scholars.The concepts of skew brace and semi-brace have mushroomed.In recent years,in order to find the non-degenerate solution of Yang-Baxter equation,Angiono,Galindo and Vendramin proposed the concept of Hopf brace on Hopf algebra for the first time,and pointed out that each Hopf brace could construct the solution of the braid equation.Recently,Huihui Zheng et al.proposed the concept of Hopf cobrace,and gave the relationship between Hopf cobrace,braid equation and bicrossed coproduct.In an attempt to study the origins and the nature of the law binding together two group operations into a skew brace.Brzezi?ski generalized skew brace.Furthermore,Brzezi?ski proposed the concept of Hopf truss,which is Hopf algebra version of skew trusses,and gave equivalent characterisations of Hopf trusses.Based on the above analysis and consideration,this paper starts from the lower order group,and constructs a brace with order not greater than 10.Moreover,we introduce the conception of Hopf cotrusses,give their equivalent characterisations,and establish the relationship between Hopf cotruss and Hopf cobrace.This thesis is divided into four chapters:In Chapter 1,we give the research background of this article,and introduce the development of brace.In Chapter 2,we introduce the theoretical basis and give the main results of this article.In Chapter 3,we give the construction of the brace with order not greater than 10.In chapter 4,we introduce the conception of Hopf cotrusses,give their equivalent characterisations,and establish the relationship between Hopf cotruss and Hopf cobrace.