Structure Group And Skew Left Brace

The quantum Yang-Baxter equation is a basic equation in mathematical physics,and is the basis of quantum group theory.In 1992,Drinfeld proposed to study the set-theoretical solution to the Yang-Baxter equation.A pair（X,S）is called a set-theoretical solution to the Yang-Baxter equation if it satisfies the braided relation:S₁₂S₂₃S₁₂=S₂₃S₁₂S₂₃,where X is a nonempty set,S:X×X→X×X is a map and S₁₂=S×id_X,S₂₃=id_X×S.In 1999,Etingof et al.systematically studied the non-degenerate involutive set-theoretical solutions to the Yang-Baxter equation.They introduced the concept of struc-tural groups.By means of structural groups they proved the equivalence of the category of the non-degenerate symmetric sets with the category of the bijective 1-cocycles.Using structural groups of non-degenerate symmetric sets they examined decomposable non-degenerate symmetric sets,and constructed linear,affine,multipermutation and other set-theoretical solutions respectively.In the paper,they also gave a classification of low-order non-degenerate involutive set-theoretic solutions.In 2000 Lu et al.studied the non-degenerate set-theoretical solutions to the Yang-Baxter equation.They construct non-degenerate solutions of the Yang-Baxter equation in terms of bijection 1-cocycles and braiding operators on a group with generators satisfying the braid relation.These two papers are the early literature on the set-theoretical solutions to the Yang-Baxter equation.In 2007,Rump discovered the relationship between Jacobson radical rings and non-degenerate involutive set-theoretical solutions to the Yang-Baxter equation.As a gener-alization of radical rings,he proposed the concept of left brace,and gave the relationship between left braces and non-degenerate involutive set-theoretic solutions to the Yang-Baxter equation.In 2017,Guarnieri et al.generalized the concept of left braces to skew left braces and gave the relationship between skew left braces and non-degenerate set-theoretical solutions to Yang-Baxter equation.Skew left braces have attracted many people to study from algebraic points of view.Among others,the additive group and the multiplicative group of a skew left brace and their interaction are important topics.In 2017,Guarnieri et al.studied finite skew left braces whose multiplicative groups are Abelian or nilpotent.In 2009,Cedóet.al.studied finite skew left braces with nilpotent additive groups.In 2019,Nasybullov studied relationship between the additive group and the multiplicative group of a finite skew two-side brace.It is a fundamental problem to characterize and classify the skew braces with cyclic additive or multiplicative groups.In2019,Rump completed the classification of the skew left braces whose additive group is an infinite cyclic group,though little is known for skew left braces whose multiplicative group is an infinite cyclic group.The paper published by Etingof et.al.in 1999 is classical for research of the set-theoretical solutions to the Yang-Baxter equation,and it is rather difficult for a beginner.The second chapter of this dissertation provides some details and annotation to the paper that would be helpful for beginners.The third chapter introduces skew left braces and their relationship with non-degenerate set-theoretical solutions.We also study skew left braces with infinite cyclic multiplicative groups.It is proved that if A is a skew left brace with an infinite cyclic multiplicative group and（A/Soc（A）,+）is cyclic,then A is a trivial brace.