Hopf Algebroids and Their Cyclic Theory
Hopf Algebroids and Their Cyclic Theory
复制标题
Hopf 代数胚及其循环理论
DOI：
--
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
N. Kowalzig
中科院分区：
文献类型：
--
作者：
N. Kowalzig
本论文的主要目的是澄清非交换几何中的广义对称性概念（即，群胚和李代数胚的非交换类似物）及其相关的（上）同调。这些想法被纳入的概念，霍普夫代数胚和霍普夫循环（上）的同源性。在现有的各种方法中，我们专注于定义的霍普夫代数体的Boehm-Szlachanyi，理解的霍普夫代数体组成的左，右bialgebroid结构和对极交织这些结构。我们详细讨论的双代数体和霍普夫代数体结构的例子包括李-莱因哈特代数的通用包络代数，喷气空间，卷积和函数代数在etale广群胚。广义Connes-Moscovici代数，即，本文还讨论了任意etale群胚上的横微分算子空间：我们给出了Connes-Moscovici和Moscovici-Rangipour构造的背景过程的一般描述，引入了双代数胚和双交叉积双代数胚的匹配对的概念。Connes-Moscovici代数就可以这样产生。进一步的一般建设，我们提供包括例如一个范畴之间的等价左bialgebroid余模和模的意义上的Kadison-Szlachanyi。在这篇论文的中心，我们指出，Hopf-循环上同调是自然的定义时，使用的概念，从Boehm-Szlachanyi的Hopf代数体：我们解释了如何的Hopf-循环上同调适合monoidal范畴的（Hopf代数体）模，并表明，它下降在一个典型的方式从循环上同调的coring。这推广了Crainic对Hopf代数的类似方法。我们还开发了一个双循环同源理论的Hopf代数体，获得循环对偶的意义上的Connes和广义的Hopf-Galois映射（在Schauenburg的意义上）规范相关联的Hopf代数体。需要这样的映射来调解所涉及的双代数体模和余代数范畴。分支的理论，我们讨论包括鉴定的Hochschild理论作为某些衍生函子。此外，我们给出了一般的结构定理的循环理论的交换和上交换的Hopf代数胚在各自的Hochschild群。这概括了类似的考虑霍普夫代数的工作Khalkhali-Rangipour。然后，我们计算了Hopf代数胚的具体例子中的Hopf-循环上同调和对偶Hopf-循环同调，如Lie-Rinehart代数的泛包络代数，etale群胚上的jet空间和卷积代数.这些计算的结果是建立Hopf-循环（上）同调作为Lie-Rinehart（上）同调和广群同调的非交换扩展。此外，还构造了一种特殊的方法来获得卷积代数的对偶Hopf-循环同调。这是通过一种方法来实现的，该方法显示了如何在特定情况下的对偶理论适合monoidal类别（左和右bialgebroid）余模。最后，我们证明了一个定理，暗示， Schauenburg意义下的imes_A$-Hopf代数是代数（上）同调理论中乘法结构（如杯、帽和Yoneda积）和某些对偶同构的关键概念。特别地，用这种方法得到了货车den Bergh关于Hochschild（co）同调和Huebschmann关于Lie-Rinehart（co）同调的结果.
The main objective of this thesis is to clarify concepts of generalised symmetries in noncommutative geometry (i.e., the noncommutative analogue of groupoids and Lie algebroids) and their associated (co)homologies. These ideas are incorporated by the notion of Hopf algebroids and Hopf-cyclic (co)homology. Among the various existing approaches, we focus on the definition of Hopf algebroids by Boehm-Szlachanyi, which understands a Hopf algebroid as consisting of left and right bialgebroid structures and an antipode intertwining these structures. Examples of bialgebroid and Hopf algebroid structures we discuss in detail include universal enveloping algebras of Lie-Rinehart algebras, jet spaces, and convolution and function algebras over etale groupoids. Generalised Connes-Moscovici algebras, i.e., spaces of transverse differential operators on arbitrary etale groupoids are also treated: we give a general description of the background procedure of the constructions by Connes-Moscovici and Moscovici-Rangipour, introducing the concept of matched pairs of bialgebroids and bicrossed product bialgebroids. The Connes-Moscovici algebras can then be shown to arise in such a way. Further general constructions we provide include for instance a categorical equivalence between left bialgebroid comodules and modules over its duals in the sense of Kadison-Szlachanyi. Central in this thesis, we indicate that Hopf-cyclic cohomology is naturally defined when using the concept of Hopf algebroids from Boehm-Szlachanyi: we explain how the Hopf-cyclic cohomology fits into the monoidal category of (Hopf algebroid) modules and show that it descends in a canonical way from the cyclic cohomology of corings. This generalises an analogous approach for Hopf algebras by Crainic. We also develop a dual cyclic homology theory for Hopf algebroids, obtained by cyclic duality in the sense of Connes and a generalised Hopf-Galois map (in the sense of Schauenburg) canonically associated to the Hopf algebroid. Such a map is required to mediate between the involved bialgebroid module and comodule categories. Ramifications of the theory we discuss comprise the identification of the Hochschild theory as certain derived functors. Also, we give general structure theorems for the cyclic theory of commutative and cocommutative Hopf algebroids in terms of their respective Hochschild groups. This generalises similar considerations for Hopf algebras in the work of Khalkhali-Rangipour. We then calculate Hopf-cyclic cohomology and dual Hopf-cyclic homology in concrete examples of Hopf algebroids, such as universal enveloping algebras of Lie-Rinehart algebras, jet spaces and convolution algebras over etale groupoids. The results of these computations are establishing Hopf-cyclic (co)homology as a noncommutative extension of both Lie-Rinehart (co)homology and groupoid homology. Moreover, a special method to obtain dual Hopf-cyclic homology for convolution algebras is constructed. This is achieved by a method which shows how in particular cases the dual theory fits into the monoidal category of (left and right bialgebroid) comodules. Finally, we prove a theorem that intimates that $ imes_A$-Hopf algebras (in the sense of Schauenburg) are a key concept for multiplicative structures (such as cup, cap and Yoneda products) and certain duality isomorphisms in algebraic (co)homology theories. In particular, results on Hochschild (co)homology by Van den Bergh and Lie-Rinehart (co)homology by Huebschmann are obtained this way.