Leibniz Pair的形式形变

发布时间：2018-01-07 20:13

本文关键词：Leibniz Pair的形式形变　出处：《安徽大学》2017年硕士论文　论文类型：学位论文

【摘要】：形变理论是代数学中重要研究内容之一,它与代数几何,代数表示论,同调代数,非交换几何,代数拓扑等领域都有着密切的关系.在研究Poisson代数的形变理论时,M.Flato,M.Gerstenhaber和A.A.Vovonov等人于1995年首先引入了Leibniz pair的定义.之后,Leibniz Pair得到了代数学家们广泛的关注.本文主要研究Leibniz Pair的几种形式形变理论.第一章,我们主要回顾结合代数和Lie代数的形式形变理论,包括这两类代数的形变方程,Hochschild上同调与Lie代数的Chevalley-Eilenberg上同调.第二章,我们首先回顾了Leibniz Pair与Leibniz Pair上同调的定义;其次,我们讨论了Leibniz Pair的整体形式形变,即Leibiniz pair中的结合代数与Lie代数同时形变;最后,我们详细证明了Leibniz Pair的整体形变正是由Leibniz Pair上同调所控制.值得指出的是,M.Flato等人已经指出这一事实,但未给出证明.第三章,我们研究了Leibniz Pair的两种单侧形变,即结合代数发生形变而Lie代数不作形变,与Lie代数发生形变而结合代数不作形变.考虑Leibniz Pair双复形的两种谱序列,我们构造了两种新的上同调群.进而通过讨论Leibniz Pair的两种单侧形变的形变方程,我们证明了这两种单侧形变由这两种新的上同调所控制.
[Abstract]:Deformation theory is one of the important research contents in algebra, which is related to algebraic geometry, algebraic representation theory, homology algebra, noncommutative geometry. In the study of deformation theory of Poisson algebra, M. Flato has a close relationship in the field of algebraic topology. In 1995, M. Gerstenhaber and A. A. Vovonov first introduced the definition of Leibniz pair. Leibniz Pair has been paid more and more attention by mathematicians. In this paper, several formal deformation theories of Leibniz Pair are studied. Chapter 1. We mainly review the formal deformation theory combined with algebra and Lie algebra, including the deformation equations of these two algebras. Hochschild cohomology and Chevalley-Eilenberg cohomology of Lie algebra. Chapter 2. We first review the definition of cohomology between Leibniz Pair and Leibniz Pair. Secondly, we discuss the global deformation of Leibniz Pair, that is, the associative algebra in Leibiniz pair and the Lie algebra at the same time. Finally, we prove in detail that the global deformation of Leibniz Pair is controlled by the cohomology of Leibniz Pair. M. Flato et al. Has pointed out this fact, but has not given the proof. In chapter 3, we study two kinds of unilateral deformation of Leibniz Pair. That is, associative algebras deform but Lie algebras do not deform, and Lie algebras deform and associative algebras do not deform. Two spectral sequences of Leibniz Pair bicomplex are considered. We construct two new cohomology groups, and by discussing the deformation equations of two unilateral deformations of Leibniz Pair, we prove that these two unilateral deformations are controlled by these two cohomology.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O15

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