發(fā)布時(shí)間：2018-04-01 11:08

  本文選題：Hom-李代數(shù)　切入點(diǎn)：Hom-Nijenhuis-Richardson括號(hào)　出處：《吉林大學(xué)》2016年博士論文

【摘要】：本文主要研究了Hom-李雙代數(shù)胚及其相關(guān)理論,特別的利用Hom-大括號(hào)對(duì)Hom-李雙代數(shù)進(jìn)行了細(xì)致的研究。經(jīng)典的大括號(hào)實(shí)質(zhì)上是定義在余切叢上的分次泊松括號(hào).它是研究李雙代數(shù)的一個(gè)非常有效的工具.目前已經(jīng)對(duì)大括號(hào)理論有了很多推廣和應(yīng)用.本文,給出了大括號(hào)在Hom-情形的形式,即Hom-大括號(hào).也就是說(shuō)給出了一個(gè)研究Hom-結(jié)構(gòu)的比較有效的工具.由于分次向量空間上的Nijenhuis-Richardson括號(hào)是大括號(hào)定義的一部分,因此我們首先定義Hom-Nijenhuis-Richardson括號(hào),并且證明Hom-Nijenhuis-Richardson括號(hào)可以給出一個(gè)分次Hom-李代數(shù)結(jié)構(gòu)Hom-Nij enhuis-Richardson括號(hào)有很多好的性質(zhì).一方面,它可以用來(lái)描述Hom-李代數(shù)結(jié)構(gòu).另一方面,還可以給出一個(gè)與Hom-李代數(shù)的表示密切相關(guān)的例子.這個(gè)例子對(duì)我們考慮Hom-李代數(shù)的相關(guān)問(wèn)題,具有非常重要的啟發(fā)意義.另外,由Hom-Nijenhuis-Richardson括號(hào),可以誘導(dǎo)一個(gè)上同調(diào)算子.這個(gè)由Hom-大括號(hào)決定的上同調(diào)算子與現(xiàn)有的上同調(diào)算子不同.也就是說(shuō)Hom-李代數(shù)的上同調(diào)理論并不唯一.接下來(lái).我們引入Hom-大括號(hào)的概念.并且證明它也可以給出一個(gè)分次Hom-李代數(shù)結(jié)構(gòu).除此之外,大括號(hào)還可以給出一個(gè)purely Hom-泊松結(jié)構(gòu).作為Hom-大括號(hào)理論的應(yīng)用.我們給出了Hom-Nijenhuis算子的定義.在[42]由Hom-Nijenhuis算子的概念首次被提出.然而,本文中所定義的Hom-Nijenhuis算子與前文中所定義的并不相同.我們還可以證明Hom-Nijenhuis算子與平凡形變一一對(duì)應(yīng),當(dāng)然這里的Hom-李代數(shù)的平凡形變與前文也不一樣,我們這里給出的平凡形變定義本身就包含了“扭曲”態(tài)射的信息在里面.同樣的,Homm-O-算子的概念也與[43]中所定義的有所不同.但是我們認(rèn)為本文中的定義方式更加合理(見(jiàn)注3.3.1和3.3.2),這恰恰說(shuō)明我們定義的Homm-大括號(hào)是一個(gè)非常有效的工具.我們還闡述了Hom-0-算子和Hom-Nijenhuis算子以及Homm-右對(duì)稱算子算子之間的聯(lián)系(引理3.3.2與命題3.3.2).利用Hom-大括號(hào)可以定義Hom-李雙代數(shù).本文給出的Hom-李雙代數(shù)的定義與文獻(xiàn)[47]中保持一致,但是這種定義方法不能給出Manin三元組理論.除此之外,Hom-李雙代數(shù)還有一種定義方式,見(jiàn)[43].但是由于后一種定義對(duì)余伴隨表示的存在性依賴較強(qiáng),要加上很強(qiáng)的限制條件.為此,我們提出一種改進(jìn)型的定義,文中稱為purely Hom-李雙代數(shù).這個(gè)新的定義,即可以成功實(shí)現(xiàn)Manin三元組理論,又沒(méi)有對(duì)余伴隨表示的依賴.跟經(jīng)典情形類似,Hom-李雙代數(shù)(VV*)是一對(duì)Hom-李代數(shù)(Kμ)與(V*,△)滿足相容性條件.此相容性條件也有三種等價(jià)的描述：上同調(diào)算子滿足導(dǎo)子性質(zhì),余乘是一階閉鏈,V(?)V*可以給出Homm-李代數(shù)結(jié)構(gòu).隨后,我們用Hom-大括號(hào)語(yǔ)言給出了Hom-Lie quasi-雙代數(shù)和Hom-quasi-Lie雙代數(shù)的定義,并給出其通常代數(shù)語(yǔ)言描述的等價(jià)定義.Homm-李代數(shù)胚的定義在[32]中首次引入.本文給出的新的Hom-李代數(shù)胚的定義與[32]有關(guān)系,但是不完全一致.我們定義了Hom-李代數(shù)胚上的上同調(diào)算子,縮并算子和李導(dǎo)數(shù)算子等微分運(yùn)算,給出了嘉當(dāng)公式等一些重要等式.證明了(A→M,φ,[·,·]A,α,(?)A)是一個(gè)Homm-李代數(shù)胚當(dāng)且僅當(dāng)((?)kΤ(∧kA*),∧,(?)A(?),d)是一個(gè)((?)A(?),(?)A(?))-微分分次交換代數(shù).進(jìn)一步,給出Hom-李雙代數(shù)胚的概念,它是李雙代數(shù)胚的推廣,證明了一個(gè)Hom-李雙代數(shù)胚的底流形上有自然的Hom-泊松代數(shù)的結(jié)構(gòu).最后,給出了Hom-Courant代數(shù)胚的概念.它是Courant代數(shù)胚的推廣.我們把李雙代數(shù)胚和Courant代數(shù)胚的一些經(jīng)典公式推廣到Homm-情形.最后,我們給出了Homm-李雙代數(shù)胚和Hom-Courant代數(shù)胚的關(guān)系：對(duì)一個(gè)Hom-李雙代數(shù)胚(A,A*).A(?)A*上有一個(gè)自然的Hom-Courant代數(shù)胚結(jié)構(gòu).
[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 緇欏嚭鍏墮，

本文編號(hào)：1695286