Hom-李双代数胚

发布时间：2018-04-01 11:08

  本文选题：Hom-李代数　切入点：Hom-Nijenhuis-Richardson括号　出处：《吉林大学》2016年博士论文

【摘要】：本文主要研究了Hom-李双代数胚及其相关理论,特别的利用Hom-大括号对Hom-李双代数进行了细致的研究。经典的大括号实质上是定义在余切丛上的分次泊松括号.它是研究李双代数的一个非常有效的工具.目前已经对大括号理论有了很多推广和应用.本文,给出了大括号在Hom-情形的形式,即Hom-大括号.也就是说给出了一个研究Hom-结构的比较有效的工具.由于分次向量空间上的Nijenhuis-Richardson括号是大括号定义的一部分,因此我们首先定义Hom-Nijenhuis-Richardson括号,并且证明Hom-Nijenhuis-Richardson括号可以给出一个分次Hom-李代数结构Hom-Nij enhuis-Richardson括号有很多好的性质.一方面,它可以用来描述Hom-李代数结构.另一方面,还可以给出一个与Hom-李代数的表示密切相关的例子.这个例子对我们考虑Hom-李代数的相关问题,具有非常重要的启发意义.另外,由Hom-Nijenhuis-Richardson括号,可以诱导一个上同调算子.这个由Hom-大括号决定的上同调算子与现有的上同调算子不同.也就是说Hom-李代数的上同调理论并不唯一.接下来.我们引入Hom-大括号的概念.并且证明它也可以给出一个分次Hom-李代数结构.除此之外,大括号还可以给出一个purely Hom-泊松结构.作为Hom-大括号理论的应用.我们给出了Hom-Nijenhuis算子的定义.在[42]由Hom-Nijenhuis算子的概念首次被提出.然而,本文中所定义的Hom-Nijenhuis算子与前文中所定义的并不相同.我们还可以证明Hom-Nijenhuis算子与平凡形变一一对应,当然这里的Hom-李代数的平凡形变与前文也不一样,我们这里给出的平凡形变定义本身就包含了“扭曲”态射的信息在里面.同样的,Homm-O-算子的概念也与[43]中所定义的有所不同.但是我们认为本文中的定义方式更加合理(见注3.3.1和3.3.2),这恰恰说明我们定义的Homm-大括号是一个非常有效的工具.我们还阐述了Hom-0-算子和Hom-Nijenhuis算子以及Homm-右对称算子算子之间的联系(引理3.3.2与命题3.3.2).利用Hom-大括号可以定义Hom-李双代数.本文给出的Hom-李双代数的定义与文献[47]中保持一致,但是这种定义方法不能给出Manin三元组理论.除此之外,Hom-李双代数还有一种定义方式,见[43].但是由于后一种定义对余伴随表示的存在性依赖较强,要加上很强的限制条件.为此,我们提出一种改进型的定义,文中称为purely Hom-李双代数.这个新的定义,即可以成功实现Manin三元组理论,又没有对余伴随表示的依赖.跟经典情形类似,Hom-李双代数(VV*)是一对Hom-李代数(Kμ)与(V*,△)满足相容性条件.此相容性条件也有三种等价的描述：上同调算子满足导子性质,余乘是一阶闭链,V(?)V*可以给出Homm-李代数结构.随后,我们用Hom-大括号语言给出了Hom-Lie quasi-双代数和Hom-quasi-Lie双代数的定义,并给出其通常代数语言描述的等价定义.Homm-李代数胚的定义在[32]中首次引入.本文给出的新的Hom-李代数胚的定义与[32]有关系,但是不完全一致.我们定义了Hom-李代数胚上的上同调算子,缩并算子和李导数算子等微分运算,给出了嘉当公式等一些重要等式.证明了(A→M,φ,[·,·]A,α,(?)A)是一个Homm-李代数胚当且仅当((?)kΤ(∧kA*),∧,(?)A(?),d)是一个((?)A(?),(?)A(?))-微分分次交换代数.进一步,给出Hom-李双代数胚的概念,它是李双代数胚的推广,证明了一个Hom-李双代数胚的底流形上有自然的Hom-泊松代数的结构.最后,给出了Hom-Courant代数胚的概念.它是Courant代数胚的推广.我们把李双代数胚和Courant代数胚的一些经典公式推广到Homm-情形.最后,我们给出了Homm-李双代数胚和Hom-Courant代数胚的关系：对一个Hom-李双代数胚(A,A*).A(?)A*上有一个自然的Hom-Courant代数胚结构.
[Abstract]:This paper mainly studies Hom- Li Shuang Algebroid and related theories, especially the use of braces Hom- makes a careful study of the Hom- algebra. Li Shuang brace essence is defined on a classical Poisson brackets in the cotangent bundle. It is a very effective tool to study Li Shuang algebras. Has a pair of braces there are a lot of theories of promotion and application. This paper gives the braces in Hom- form, namely Hom- braces. That gives a more effective tool for a study of the Hom- structure. Because the graded vector space Nijenhuis-Richardson is a part of the braces are defined, so we first define Hom-Nijenhuis-Richardson in parentheses, and prove that the Hom-Nijenhuis-Richardson bracket can be given a graded Lie algebra Hom- structure Hom-Nij enhuis-Richardson bracket has many nice properties. On the one hand, it can Hom- is used to describe the lie algebra structure. On the other hand, can also give a Hom- representation of Lie algebra is closely related to the example. This example for us to consider issues related to the Hom- algebra, has very important significance. In addition, the Hom-Nijenhuis-Richardson bracket can be induced to a coherent operator. This is Hom- the braces decided the cohomology and cohomology operators of different existing operators. That is to say Hom- Lie algebra cohomology theory is not unique. Next, we introduce the concept of Hom- brace. And prove that it can be given a graded Hom- algebra structure. In addition, the braces can also give a purely Hom-. Application of Hom- as a Poisson structure brace theory. We give the definition of the Hom-Nijenhuis operator in [42]. By the concept of Hom-Nijenhuis operator was put forward for the first time. However, the definition of the The Hom-Nijenhuis operator is defined and the above is not the same. We can also show that the Hom-Nijenhuis operator and the ordinary deformation correspondence, of course here Hom- Lie algebra ordinary deformation and above are not the same, here we give the definition of ordinary deformation in itself contains a "twisted" morphism information inside. Similarly, the concept of Homm-O- the operator and [43] have defined different. But we think that the definition of a more reasonable way (see 3.3.1 and 3.3.2), which indicates that the Homm- brace our definition is a very effective tool. We also introduced between the Hom-0- operator and Hom-Nijenhuis operator and Homm- Operator right symmetric operator contact (lemma 3.3.2 and proposition 3.3.2). Using Hom- braces can define Hom- Li Shuang algebra. Consistent definition and literature [47] Hom- presented in the Li Shuang algebra, But this definition cannot give Manin three element theory. In addition, there is a Li Shuang algebra Hom- definition, see [43]. but because after a definition of coadjoint indicates the existence of a strong dependence, to limit the conditions of strong plus. Therefore, we propose an improved definition of the Li Shuang called the purely Hom- algebra. This new definition, which can be successfully achieved Manin three element theory, and no more than the dependent adjoint representation. With the classical case is similar to that of Hom- (VV*) is a Li Shuang algebra of Lie algebra Hom- (K) and (V*, a) satisfy the compatibility conditions. The compatibility there are three kinds of equivalent condition description: cohomology operators satisfy derivations by nature, more than a closed chain order, V (?) V* can give Homm- Lie algebra. Then, we define Hom-Lie quasi- algebra and Hom-quasi-Lie algebra of the double double braces are given by using Hom- language, and 缁欏嚭鍏堕，

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