分裂的Hom型李（超）代数的单性与两类李代数上的可积系及其Hamilton结构

发布时间：2018-06-09 01:06

本文选题：Hom-李超代数 + Hom-李代数　；参考：《哈尔滨工业大学》2017年博士论文

【摘要】：近年来,随着数学和物理的不断发展,人们开始研究Hom型李(超)代数。我们知道,Hom-李(超)代数本身就是李(超)代数的某种形变,当Hom-李(超)代数的扭曲映射为恒等映射时,Hom-李(超)代数就退化为原来的李(超)代数,所以Hom-李(超)代数可以看作是李(超)代数的推广。分解和单性是李理论中两个重要的研究内容,对于Hom型李(超)代数也可在这些方面进行研究。孤立子理论是非线性科学的研究主体之一,可积系统以及可积系统是否具有Hamilton结构也是非线性科学研究的主流方向。利用李代数的结构建立孤立子可积系统,以及扩充原有的可积系统并且得到其Hamilton结构是孤立子理论中重要的研究课题。本文一方面研究了分裂的对合的正则Hom-李代数和三类分裂的正则Hom型李超代数的分解和单性;另一方面扩充了两类李代数上的孤立子可积系,得到了孤立子可积系的双可积耦合和三可积耦合及其Hamilton结构。本文的主要内容分为三部分:第一,研究了分裂的对合的正则Hom-李代数的分解和单性。首先,定义了分裂的对合的正则Hom-李代数和它的根连通。利用根连通的性质,得到了具有对称根系的分裂的对合的正则Hom-李代数分解成若干理想的直和的充分条件。其次,得到了具有对称根系的分裂的对合的正则Hom-李代数是单的充分必要条件。最后,得到了具有对称根系的分裂的对合的正则Hom-李代数分解成若干单理想的直和的充分条件。第二,研究了分裂的正则Hom-李超代数,分裂的正则δ-Hom-Jordan李超代数和分裂的正则BiHom-李超代数的分解和单性。首先,定义了分裂的正则Hom-李超代数和它的根连通。利用根连通的性质,刻画了具有对称根系的分裂的正则Hom-李超代数分解成若干理想的直和的充分条件。并且得到了具有对称根系的分裂的正则Hom-李超代数是单的充分必要条件和具有对称根系的分裂的正则Hom-李超代数分解成若干单理想的直和的充分条件。其次,给出了分裂的正则δ-Hom-Jordan李超代数的定义和它的根连通。利用其根连通的性质,刻画了具有对称根系的分裂的正则δ-Hom-Jordan李超代数分解成若干理想的直和的充分条件。并且刻画了具有对称根系的分裂的正则δ-Hom-Jordan李超代数是单的充分必要条件和具有对称根系的分裂的正则δ-Hom-Jordan李超代数分解成若干单理想的直和的充分条件。最后,定义了分裂的正则BiHom-李超代数和它的根连通。利用根连通的性质,刻画了具有对称根系的分裂的正则BiHom-李超代数分解成若干理想的直和的充分条件。并且刻画了具有对称根系的分裂的正则BiHom-李超代数是单的充分必要条件和具有对称根系的分裂的正则BiHom-李超代数分解成若干单理想的直和的充分条件。第三,研究了李代数S O(3)和S O(4)上的孤立子可积系的双可积耦合和三可积耦合及其Hamilton结构。首先,利用三维李代数S O(3)上的孤立子可积系,从它的扩展的谱矩阵和扩展的零曲率方程得到双可积耦合和三可积耦合。然后由迹恒等式得到双可积耦合和三可积耦合相应的Hamilton结构。其次,利用六维李代数S O(4)上的的孤立子可积系,从它的扩展的谱矩阵和扩展的零曲率方程得到双可积耦合和三可积耦合。然后由迹恒等式得到双可积耦合和三可积耦合相应的Hamilton结构。
[Abstract]:In recent years, with the continuous development of mathematics and physics, people have begun to study Hom type lie (superalgebra) algebra. We know that Hom- lie (super) algebra itself is a kind of deformation of lie (superalgebra) algebra. When the distortion mapping of Hom- lie (super) algebra is a identity mapping, Hom- lie (super) algebra will degenerate into the original lie (super) algebra, so Hom- lie (super) algebra can As the extension of Li (super) algebra, decomposition and monasexual are two important research contents in Li's theory. The Hom type lie (super) algebra can also be studied in these aspects. Soliton theory is one of the main subjects of nonlinear science. The Hamilton structure of integrable systems and integrable systems is also the mainstream of nonlinear scientific research. Direction. The construction of the soliton integrable system with the structure of Lie algebra, and the expansion of the original integrable system and its Hamilton structure are important research topics in the soliton theory. On the one hand, this paper studies the decomposition and mononexy of the split pair regular Hom- Lie algebra and the three class of regular Hom type Li Chao algebras. This paper extends the soliton integrable system on two classes of Lie algebras and obtains the double integrable and three integrable coupling and Hamilton structures of the integrable integrable systems of the soliton. The main contents of this paper are divided into three parts. First, the decomposition and mononexitarisk of the split regular Hom- Lie algebra are studied. First, the split regular Hom- Lie algebra is defined. The root connectivity is connected. Using the property of the root connectivity, we get the sufficient conditions for the decomposition of the split regular Hom- Lie algebra with the symmetric root system into some ideal direct sum conditions. Secondly, we get the necessary and sufficient conditions for the regular Hom- Lie algebra with the split of the symmetric root system. Finally, the splitting of the symmetric root is obtained. The involution regular Hom- Lie algebra is decomposed into some sufficient conditions for single ideal straight sum. Second, the splitting of regular Hom- Li Chao algebra, split regular Delta -Hom-Jordan Li Chao algebra and split regular BiHom- Li Chao algebra are studied. First, the split regular Hom- Li Chao algebra and its root connectedness are defined. In this connection, we characterize the decomposition of the regular Hom- Li Chao Algebra with the split of the symmetric roots into some ideal direct sum conditions. And the regular Hom- Algebra with the splitting of the symmetric root is the sufficient and necessary condition of the single and the positive Hom- Li Chao Algebra with the split of the symmetric root system into some single ideal. Secondly, the definition of the split regular Delta -Hom-Jordan Li Chao algebra and its root connectedness are given. Using the properties of its root connectedness, we characterize the sufficient conditions for the decomposition of the regular Delta -Hom-Jordan Algebra with the splitting of the symmetric root system into some ideal direct sum conditions, and describe the positive splitting of the symmetric root. Then the delta -Hom-Jordan Li Chao algebra is a sufficient and necessary condition of the single and a regular Delta -Hom-Jordan Li Chao Algebra with the split of the symmetric root system into a number of single ideal direct sum conditions. Finally, the split regular BiHom- Li Chao algebra and its root connectedness are defined. The splitting of the root is characterized by the properties of the root connected. The regular BiHom- Li Chao algebra is decomposed into some ideal direct sum sufficient conditions. And the regular BiHom- Li Chao Algebra with the splitting of the symmetric root is a sufficient and necessary condition of the single and the regular BiHom- Li Chao Algebra with the split of the symmetric root is decomposed into a number of single ideal direct sum conditions. Third, the study of Li Dai The double integrable coupling and three integrable coupling and its Hamilton structure on the integrable system on the S O (3) and S O (4). First, the integrable coupling and the three integrable coupling are obtained from the extended spectral matrix and the extended zero curvature equation by the isolated subintegrable system on the three-dimensional Lie algebra S O (3). Then the double integrable coupling is obtained by the trace identity and the double integrable coupling is obtained by the trace identity. Three the corresponding Hamilton structure can be integrable coupled. Secondly, using the soliton integrable system on the six dimensional Lie algebra S O (4), the double integrable coupling and the three integrable coupling are obtained from its extended spectral matrix and the extended zero curvature equation. Then the Hamilton structure with the dual integrable coupling and the three integrable coupling is obtained by the trace identity.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O152.5

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