可遞李代數(shù)胚分類空間的研究
發(fā)布時(shí)間：2019-01-24 19:46
【摘要】：李代數(shù)胚是李代數(shù)與流形切叢的推廣,它在Poisson幾何和非交換幾何中有大量的應(yīng)用�？蛇f李代數(shù)胚是它的一個(gè)重要分支,是該領(lǐng)域的主要研究?jī)?nèi)容之一。本文從李代數(shù)叢入手,研究了李代數(shù)叢經(jīng)由切叢擴(kuò)張為可遞李代數(shù)胚的相關(guān)問(wèn)題、可遞李代數(shù)胚拉回的同倫不變性、和可遞李代數(shù)胚的分類空間,同時(shí)討論了可遞李代數(shù)胚的范疇示性類。所得主要結(jié)果如下:首先,研究了李代數(shù)叢與切叢間的耦合。Mackenzie在研究李代數(shù)胚的擴(kuò)張問(wèn)題時(shí)引入了耦合的定義,說(shuō)明了李代數(shù)叢可經(jīng)由切叢擴(kuò)張為可遞李代數(shù)胚的必要條件是切叢與李代數(shù)叢之間存在耦合。本文給出了判定耦合存在性的充分必要條件,然后定義了耦合之間的等價(jià)關(guān)系,并且利用從底流形到一個(gè)特定分類空間的連續(xù)映射的同倫等價(jià)類來(lái)描述耦合的等價(jià)類。其次,說(shuō)明了用于判定李代數(shù)叢可否經(jīng)由切叢擴(kuò)張成可遞李代數(shù)胚的Mackenzie阻礙類具有函子性質(zhì)。對(duì)于單連通流形上的李代數(shù)叢及其耦合所對(duì)應(yīng)Mackenzie阻礙類構(gòu)造了具有萬(wàn)有性質(zhì)的上同調(diào)元素。證明了當(dāng)李代數(shù)叢的底空間是單連通流形時(shí),Mackenzie阻礙類是平凡的,即它是上同調(diào)群中的零元素。對(duì)于底空間沒(méi)有限制條件的情況下,證明了當(dāng)李代數(shù)叢的纖維是可約李代數(shù)時(shí),其Mackenzie阻礙類也是平凡的。然后,證明了可遞李代數(shù)胚的拉回具有同倫不變性,建立了從光滑流形范疇到可遞李代數(shù)胚范疇的同倫函子,討論了之前所得到的關(guān)于耦合與Mackenzie阻礙類的成果對(duì)于研究可遞李代數(shù)胚分類空間的重要作用。說(shuō)明可以通過(guò)函子間的自然變換來(lái)定義可遞李代數(shù)胚的示性類,并且在伴隨叢是交換李代數(shù)叢的可遞李代數(shù)胚范疇內(nèi)定義了一系列示性類,然后與Kubarski推廣的李代數(shù)胚的Chern-Weil同態(tài)所定義的示性類作對(duì)比,說(shuō)明該Chern-Weil同態(tài)并不能構(gòu)造所有可遞李代數(shù)胚的示性類。
[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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：O152.5
，
本文編號(hào)：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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：O152.5
，
本文編號(hào)：2414784
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