Abstract

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
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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 $\times_A$-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.
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