Set-Theoretical Solutions To Hom-Yang-Baxter Equation And Relevant Algebraic Structures

Let V be a vector space,R:V(?)V→V(?)V be a linear map.If R satisfies the following equation:(R(?)id)(id(?)R)(R(?)id)=(id(?)R)(R(?)id)(id(?)R), then R is called a solution to the Yang-Baxter equation.There are several types of Yang-Baxter equations and corresponding algebraic structures,such as the(quantum)Yang-Baxter equation and quantum groups,the dynamical quantum Yang-Baxter equation and dynamical quantum groups,the(quantum)Hom-Yang-Baxter equation and Hom-quantum groups,the classical Yang-Baxter equation and Lie algebras,the classical Hom-Yang-Baxter equation and Hom-Lie algebras,the associative Yang-Baxter equation and Rota-Baxter algebras,and so on.In order to find more solutions to the Yang-Baxter equation,Drinfeld firstly proposed set-theoretic solutions to the Yang-Baxter equation in 1992.Let X be a set and r:X×X→X×X be a map.If r satisfies the following equation:(r×id)(id×r)(r×id)=(id×r)(r×id)(id×r),then r is called a set-theoretic solution to the Yang-Baxter equation.Solutions to the Yang-Baxter equation can be constructed by using set-theoretic solutions.At the end of the 20th century,Gateva-Ivanova and Van den Bergh,as well as Etingof et al.began studying a special class of set-theoretic solutions—non-degenerate involutive solutions.Rump introduced an algebraic structure called a cycle set,which is a left quasigroup satisfying(xy)(xz)=(yx)(yz)for any x,y∈X.Rump proved that there is a bijective correspondence between cycle sets and left non-degenerate involutive set-theoretic solutions to the Yang-Baxter equation,and further introduced the left braces.Left braces can be used to construct non-degenerate involutive set-theoretic solutions,and vice versa.Guarnieri and Vendramin introduced generalized left braces—called skew left braces,which can be used to construct non-degenerate set-theoretic solutions.Brace and skew left brace have now become a very active research field in algebra.In 1984,Gervais and Neveu first proposed the dynamical Yang-Baxter equation.In order to study such equations,Etingof developed the theory of dynamical quantum groups.In 2005,Shibukawa proposed set-theoretic solutions to the dynamical Yang-Baxter equation.Then,new algebraic structures have been introduced,such as dynamical cycle sets,dynamical braces,etc.In 2009,Yau proposed the Hom-Yang-Baxter equation inspired by Hom-Lie alge-bras.Let V be a vector space,R:V(?)V→V(?)V andα:V→V be two linear maps.If R satisfies R(α(?)α)=(α(?)α)R and (R(?)α)(α(?)R)(R(?)α)=(α(?)R)(R(?)α)(α(?)R),then R is called a solution to the Hom-Yang-Baxter equation.The study of such equation has led to the theory of Hom-quantum groups.The quantum Hom-Yang Baxter equation and the classical Hom-Yang Baxter equation have attracted many author’s attentions,but the set-theoretic solutions to the Hom-Yang Baxter equation have not been studied yet.This dissertation studies set-theoretic solutions to the Hom-Yang-Baxter equation and related algebraic structures.In chapter 2,we firstly introduce the concept of set-theoretic solutions to Hom-Yang-Baxter equation.Let X be a set,r:X×X→X×X andα:X→X be maps.If r satisfies r(α×α)=(α×α)r and (r×α)(α×r)(r×α)=(α×r)(r×α)(α×r), then r is called a set-theoretic solution to the Hom-Yang-Baxter equation.Using set-theoretic solutions we can construct solutions to the Hom-Yang-Baxter equation.Fur-thermore,we characterize left non-degenerate involutive set-theoretic solutions to the Hom-Yang-Baxter equation.Then,we introduce a new algebraic structure—Hom-cycle sets,and establish a one-to-one correspondence between Hom-cycle sets and left non-degenerate involutive set-theoretic solutions to the Hom-Yang-Baxter equation.Finally,we discuss the relationship between Hom-cycle sets and cycle sets;As an application,we present relationship between set-theoretic solutions to the Hom-Yang-Baxter equation and set-theoretic solutions to the Yang-Baxter equation.In order to study non-involutive set-theoretic solutions to the Hom-Yang-Baxter equation,in Chapter 3,we introduce a class of left non-degenerate set-theoretic solutions to the Hom-Yang-Baxter equation,called I-type set-theoretic solutions,that is,solutions that satisfy r~2=α×α.We provide a characterization of a left non-degenerate I-type set-theoretic solutions.Furthermore,we introduced I-type Hom-cycle sets and established a one-to-one correspondence between I-type Hom-cycle sets and left non-degenerate I-type set-theoretic solutions to the Hom-Yang-Baxter equation.In order to study general left non-degenerate set-theoretic solutions to the Hom-Yang-Baxter equation,in Chapter 4,we introduce the concept of the Hom-q-cycle set,es-tablish a one-to-one correspondence between the Hom-q-cycle sets and left non-degenerate set-theoretic solutions to the Hom-Yang-Baxter equation,and discusse the dynamical ex-tension of Hom-q-cycle sets.In 2018,Smoktunowicz et al.proved that the category of skew left braces is isomor-phic to the category of skew left linear cycle sets.In 2023,Bai Chengming and others proved that the category of skew left braces is isomorphic to the category of post-groups.In Chapter 5,we introduce Hom-skew left braces,Hom-skew left linear cycle sets,Hom-post-groups,and prove that the corresponding categories are isomorphic to each other.Braces are a generalization of Jacobson radical rings,and many people use method of ring theory to study braces.Lau proved that if the operation a*b=a。b-a-b on the left brace(A,+,。)is associative,then(A,+,。)is a two-side brace.In Chapter 6,we introduce the associator and distributor in left braces,and we obtained Lau’s result by using the associator of left braces.We also extend the Kegel theorem in ring theory(solvable maximal subrings are ideals)to two-side braces.It is proved that if M is a maximal subbrace of the two-side brace A and M is Baer radical brace,then M is an ideal of A,and an example of left braces is given to show Kegel theorem is invalid for left braces.