【作者】 张毅；

【导师】 罗彦锋；

【作者基本信息】 兰州大学 ， 数学·基础数学， 2020， 博士

【摘要】 带权无穷小双代数是带权结合经典杨巴方程的代数抽象,它在数学和数学物理领域扮演着重要的角色.本文对带权无穷小双代数进行了系统地研究.详言之,本文研究了带权无穷小双代数的基本性质,构造了一些经典结合代数上的带权无穷小双代数,并探讨了它与带算子代数,预李代数之间的联系.全文共分八章.第一章介绍了带权无穷小双代数的研究背景,研究动机和研究进展.为了本文的完整性,本章还回顾了本文所用到的一些基本概念和事实.第二章首先回顾了带权无穷小(单位)双代数的概念,它同时推广了 Joni和Rota提出的无穷小双代数以及Loday和Ronco提出的无穷小双代数.其次通过例子展示了一些经典的结合代数具有带权无穷小(单位)双代数结构.最后研究了带权无穷小(单位)双代数的一些基本性质.第三章研究了两种观点下的无穷小Hopf代数—Aguiar观点下的无穷小Hopf代数和Loday-Ronco观点下的无穷小Hopf代数.第四章探究了带权结合杨巴方程的解与带权无穷小双代数之间的关系.构造了矩阵代数上带权结合杨巴方程的解到带权罗巴算子的一个双射.最后引入了带权拟三角无穷小单位双代数的概念,并证明了任意一个带权拟三角无穷小单位双代数都可以诱导出一个叶形代数结构.第五章引入了带权无穷小(单位)Hopf模的概念.证明了任意A模都有一个自然的带权无穷小单位Hopf模结构,其中A是带权拟三角无穷小单位双代数.第六章提供了一个新的方法对平面根森林进行装饰.在装饰根森林上构造了一个新的余乘使其成为一个权为零的无穷小单位双代数,并证明它装配上一族嫁接算子是权为零的自由多重1-余循环无穷小单位双代数.利用森林双理想,给出了余乘的组合解释.作为应用,得到了不带装饰的根森林上的无穷小单位双代数范畴的起始对象,这些对象恰好是非交换观点下的经典Connes-Kreimer Hopf代数的研究对象.第七章首先从Hochschild上同调的对偶诱导出对称1-余循环条件.借助于这个条件,在根森林上构造了带权无穷小单位双代数.其次从带算子代数的观点去理解带权无穷小单位双代数,并自然地引入了带权多重余循环无穷小双代数的概念.最后在根森林上构造了 Loday-Ronco观点下的无穷小Hopf代数.第八章分别从带权无穷小双代数和交换带权无穷小双代数诱导了两个预李代数.第二个构造推广了 Novikov代数上的Gel’fand-Dorfman定理.作为应用,在结合代数上构造了一个预李代数结构和一个新的李代数结构.最后在装饰平面根森林上构造了新的预李代数.更多还原

【Abstract】 The concept of weighted infinitesimal bialgebras is an algebraic meaning of the nonhomogenous associative Yang-Baxter equation,which plays an important role in mathematics and mathematical physics.In this thesis,we mainly study the weighted infinitesimal bialgebra.More precisely,we investigate the basic properties of a weighted infinitesimal bialgebra and show some examples of this algebra.We explore the relationships between weighted infinitesimal bialgebras,operated algebras and pre-Lie algebras.This thesis contains eight chapters.In Chapter 1,we introduce the backgrounds,motivations and the recent developments of weighted infinitesimal bialgebras.For the completeness,we also recall some relative definitions and facts for this paper.In Chapter 2,we first recall the concept of weighted infinitesimal(unitary)bialgebras,which generalizes simultaneously the one introduced by Joni and Rota and the one initiated by Loday and Ronco.We then show that some well-known algebras possess a weighted infinitesimal(unitary)bialgebra.Some basic properties of weighted infinitesimal(unitary)bialgebras are also investigated.In Chapter 3,we study two versions of infinitesimal Hopf algebras—the infinitesimal Hopf algebras in the sense of Aguiar and the infinitesimal unitary counitary Hopf algebra in the view of Loday and Ronco.In Chapter 4,we explore the relationship between solutions of weighted AYBEs and weighted infinitesimal unitary bialgebras.We give a bi.jection between solutions of the associative Yang-Baxter equation of weight λ and Rota-Baxter operators of weight λ on matrix algebras.Finally,We show that any weighted quasitriangular infinitesimal unitary bialgebra has a dendriform algebraic structure.In Chapter 5,we introduce the concept of weighted infinitesimal Hopf modules and show that any module carries a natural structure of weighted infinitesimal unitary Hopf module over a weighted quasitriangular infinitesimal unitary bialgebra.In Chapter 6,we decorate planar rooted forests in a new way,and prove that the space of decorated planar forests,together with a coproduct and a set of grafting operations,is the free Ω-cocycle infinitesimal unitary bialgebra(resp.Hopf algebra)of weight zero on a set.A combinatorial description of the coproduct is also given.As applications,we obtain the initial object in the category of cocycle infinitesimal unitary bialgebras(resp.Hopf algebras)on undecorated planar rooted forests,which is the object studied in the(noncommutative)Connes-Kreimer Hopf algebra.In Chapter 7,we introduce the concept of symmetric 1-cocycle conditions which is derived from a dual of the Hochschild cohomology.We study the universal properties of the space of decorated planar rooted forests in the framework of operated algebras,leading to the notation of a weighted Ω-cocycle infinitesimal unitary bialgebra.We also construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco.In Chapter 8,we derive two pre-Lie algebras from an arbitrary weighted infinitesimal bialgebra and a weighted commutative infinitesimal bialgebra,respectively.The second construction generalizes the Gel’fand-Dorfman Theorem on Novikov algebras.As an application,a pre-Lie algebraic structure and then a new Lie algebraic structure on an associative algebra are constructed.Finally,we construct two new pre-Lie algebras on decorated planar rooted forests.更多还原