可递李代数胚分类空间的研究
发布时间：2019-01-24 19:46
【摘要】：李代数胚是李代数与流形切丛的推广,它在Poisson几何和非交换几何中有大量的应用。可递李代数胚是它的一个重要分支,是该领域的主要研究内容之一。本文从李代数丛入手,研究了李代数丛经由切丛扩张为可递李代数胚的相关问题、可递李代数胚拉回的同伦不变性、和可递李代数胚的分类空间,同时讨论了可递李代数胚的范畴示性类。所得主要结果如下:首先,研究了李代数丛与切丛间的耦合。Mackenzie在研究李代数胚的扩张问题时引入了耦合的定义,说明了李代数丛可经由切丛扩张为可递李代数胚的必要条件是切丛与李代数丛之间存在耦合。本文给出了判定耦合存在性的充分必要条件,然后定义了耦合之间的等价关系,并且利用从底流形到一个特定分类空间的连续映射的同伦等价类来描述耦合的等价类。其次,说明了用于判定李代数丛可否经由切丛扩张成可递李代数胚的Mackenzie阻碍类具有函子性质。对于单连通流形上的李代数丛及其耦合所对应Mackenzie阻碍类构造了具有万有性质的上同调元素。证明了当李代数丛的底空间是单连通流形时,Mackenzie阻碍类是平凡的,即它是上同调群中的零元素。对于底空间没有限制条件的情况下,证明了当李代数丛的纤维是可约李代数时,其Mackenzie阻碍类也是平凡的。然后,证明了可递李代数胚的拉回具有同伦不变性,建立了从光滑流形范畴到可递李代数胚范畴的同伦函子,讨论了之前所得到的关于耦合与Mackenzie阻碍类的成果对于研究可递李代数胚分类空间的重要作用。说明可以通过函子间的自然变换来定义可递李代数胚的示性类,并且在伴随丛是交换李代数丛的可递李代数胚范畴内定义了一系列示性类,然后与Kubarski推广的李代数胚的Chern-Weil同态所定义的示性类作对比,说明该Chern-Weil同态并不能构造所有可递李代数胚的示性类。
[Abstract]:Lie algebra is a generalization of lie algebra and manifold tangent bundle. It has a lot of applications in Poisson geometry and noncommutative geometry. Transitive lie algebra is an important branch of it and is one of the main research contents in this field. In this paper, we study the problems of the extension of lie algebras to transitive lie algebras by tangent algebras, the homotopy invariance of retractable lie algebras, and the classification spaces of transitive lie algebras. At the same time, the category representation classes of transitive lie algebras are discussed. The main results are as follows: firstly, the coupling between lie algebraic bundle and tangent bundle is studied. Mackenzie introduces the definition of coupling when studying the extension of lie algebraic germ. It is shown that the necessary condition for a lie algebraic bundle to be a transitive lie algebra is that there is a coupling between the tangent bundle and the lie algebraic bundle. In this paper, we give a necessary and sufficient condition for determining the existence of coupling, then define the equivalent relation between coupling, and use the homotopy equivalence class of continuous mapping from bottom manifold to a particular classification space to describe the equivalent class of coupling. Secondly, it is shown that the Mackenzie barrier class used to determine whether lie algebraic bundle can be extended into transitive lie algebra by tangent bundle has the property of functor. Cohomology elements with universal properties are constructed for the lie algebraic bundle on a simple connected manifold and its corresponding Mackenzie hindrance class. It is proved that when the base space of lie algebraic bundle is a simple connected manifold, the Mackenzie barrier class is trivial, that is, it is a zero element in cohomology group. It is proved that when the fiber of a lie algebra bundle is a reducible lie algebra, the Mackenzie barrier class is also trivial. Then, it is proved that the pulling back of a transitive lie algebra is homotopy invariant, and a homotopy functor from the category of smooth manifold to the category of germ of transitive lie algebra is established. In this paper, we discuss the importance of the previous results on coupling and Mackenzie obstructions to the study of the classification spaces of transitive lie algebras. It is shown that the indicative classes of transitive lie algebraic germs can be defined by natural transformations between functors, and a series of representational classes are defined in the category of transitive lie algebraic germs in which the adjoint bundle is a commutative lie algebraic bundle. Then compared with the Chern-Weil homomorphism defined by Chern-Weil of a lie algebra generalized by Kubarski, it is shown that the Chern-Weil homomorphism can not construct the representative classes of all transitive lie algebras.
【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】：2015
【分类号】：O152.5
，
本文编号：2414784
[Abstract]:Lie algebra is a generalization of lie algebra and manifold tangent bundle. It has a lot of applications in Poisson geometry and noncommutative geometry. Transitive lie algebra is an important branch of it and is one of the main research contents in this field. In this paper, we study the problems of the extension of lie algebras to transitive lie algebras by tangent algebras, the homotopy invariance of retractable lie algebras, and the classification spaces of transitive lie algebras. At the same time, the category representation classes of transitive lie algebras are discussed. The main results are as follows: firstly, the coupling between lie algebraic bundle and tangent bundle is studied. Mackenzie introduces the definition of coupling when studying the extension of lie algebraic germ. It is shown that the necessary condition for a lie algebraic bundle to be a transitive lie algebra is that there is a coupling between the tangent bundle and the lie algebraic bundle. In this paper, we give a necessary and sufficient condition for determining the existence of coupling, then define the equivalent relation between coupling, and use the homotopy equivalence class of continuous mapping from bottom manifold to a particular classification space to describe the equivalent class of coupling. Secondly, it is shown that the Mackenzie barrier class used to determine whether lie algebraic bundle can be extended into transitive lie algebra by tangent bundle has the property of functor. Cohomology elements with universal properties are constructed for the lie algebraic bundle on a simple connected manifold and its corresponding Mackenzie hindrance class. It is proved that when the base space of lie algebraic bundle is a simple connected manifold, the Mackenzie barrier class is trivial, that is, it is a zero element in cohomology group. It is proved that when the fiber of a lie algebra bundle is a reducible lie algebra, the Mackenzie barrier class is also trivial. Then, it is proved that the pulling back of a transitive lie algebra is homotopy invariant, and a homotopy functor from the category of smooth manifold to the category of germ of transitive lie algebra is established. In this paper, we discuss the importance of the previous results on coupling and Mackenzie obstructions to the study of the classification spaces of transitive lie algebras. It is shown that the indicative classes of transitive lie algebraic germs can be defined by natural transformations between functors, and a series of representational classes are defined in the category of transitive lie algebraic germs in which the adjoint bundle is a commutative lie algebraic bundle. Then compared with the Chern-Weil homomorphism defined by Chern-Weil of a lie algebra generalized by Kubarski, it is shown that the Chern-Weil homomorphism can not construct the representative classes of all transitive lie algebras.
【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】：2015
【分类号】：O152.5
，
本文编号：2414784
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