Geometric construction of Hopf cyclic characteristic classes

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.

Abstract. In this paper, we show that, under natural homological conditions, Hopf cyclichomology theory has excision. Mathematics Subject Classifications (2000): 16W30, 19D55, 57T05, 58B34. Key words: cyclic homology, Hopf algebras. 1. Introduction Cyclic cohomology theory for Hopf algebras was introduced by Connesand Moscovici in their ground breaking paper on transverse index theory[4] (cf. also [3, 5, 6] for a recent survey and new results). This theory can beregarded as the right noncommutative analogue of Lie algebra homologybecause of existence of a characteristic map for actions of Hopf algebrasand since it reduces to Lie algebra homology for Hopf algebras associ-ated to Lie algebras. This should be compared with the role played byordinary cyclic cohomology of noncommutative algebras and its reductionto de Rham homology for smooth commutative algebras [2]. The alge-braic underpinnings of Hopf cyclic theory however turned out to be morecomplicated. It is now understood as a special case of a general theorydeveloped in the series of papers [11, 12, 15, 16] (with the help of [1])for (co)algebras endowed with a (co)action of a Hopf algebra and where,among other things, the right notion of coefficients was introduced (cf. also[14] for a recent survey). This theory was later extended by the first author[13] to bialgebras and to a much larger class of coefficients. This extensionis rather surprising since existence of a bijective antipode was essential inconstructing the cyclic and cocyclic complexes developed for Hopf cyclichomology of both variants. By bialgebra or Hopf cyclic homology we nowmean the theory expounded in these papers mentioned above.The main results of the present paper are two excision theorems(Theorems 4.13 and 7.7) for Hopf cyclic homology of module coal-gebras and comodule algebras, respectively. A basic property of cyclic(co)homology for algebras, and coalgebras is excision. If by cyclic theory

In this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

One aim of this chapter is to present some standard facts about connections and their relation to characteristic classes, and then show how these ideas extend to give new kinds of characteristic classes associated to a foliation. The presentation of the material will not be at all standard, however. Indeed, the same characteristic classes associated to a foliation of codimension q can be obtained from the algebra of formal power series vector fields at the origin in ℝ q , without any reference to foliations at all. This result is consistent with the point of view that foliations are the natural quotient objects in differential geometry, but it also suggests an exposition in which higher order derivatives are incorporated from the beginning, in order to make the relation appear as clear and natural as possible. Thus, the first section is a discussion of the Lie groups of invertible polynomial mappings. The composition of two mappings of degree k is made into a mapping of degree k by discarding higher order terms, a procedure consistent with the application to Taylor series of differentiable mappings discussed in the second section. The higher order frame bundles on a manifold are principal bundles whose groups are truncated polynomial groups. In addition, each carries a Lie algebra valued 1-form, the tautological form, which contains precisely the information that the derivatives of various orders are not independent when more than one point is involved. Any G-structure has associated to it reductions of the higher order frame bundles to subgroups of the truncated polynomial groups.

On the van Est analogy in Hopf cyclic cohomology

This paper is an introduction to Hopf cyclic cohomology with an emphasis on its most recent developments. We cover three major areas: the original definition of Hopf cyclic cohomology by Connes and Moscovici as an outgrowth of their study of transverse index theory on foliated manifolds, the introduction of Hopf cyclic cohomology with coefficients by Hajac-Khalkhali-Rangipour-Sommerhäuser, and finally the latest episode on unifying the coefficients as well as extending the notion to more general settings beyond Hopf algebras. In particular, the last section discusses the relative Hopf cyclic theory that arises in the braided monoidal category settings. Contents 14 8. Rigid braided categories 21 References 24

An augmented rack is a set with a self-distributive binary operation induced by a group action, and has been extensively used in knot theory. Solutions to the Yang–Baxter equation (YBE) have been also used for knots, since the discovery of the Jones polynomial. In this paper, an interpretation of augmented racks in tensor categories is given for coalgebras that are Hopf algebra modules, and associated solutions to the YBE are constructed. Explicit constructions are given using quantum heaps and the adjoint of Hopf algebras. Furthermore, an inductive construction of Yang–Baxter solutions is given by means of the categorical augmented racks, yielding infinite families of solutions. Constructions of braided monoidal categories are also provided using categorical augmented racks.

This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms. The famous Erlanger Programm of Klein says that is about to study the transfor- mation groups of various spaces, or more precisely the properties invariant under the actions of such groups, i.e., the symmetries. Noncommutative geometry(NCG), which originated from Connes' study in operator alge- bras in 1970's, brought the landscape of many new objects and some astonishing phenomena. Back to the early 1990s, Connes and Moscovici pointed out that in noncommutative geome- try(NCG) while noncommutative spaces are represented by the algebras (usually noncommuta- tive C ∗ -algebras) of continuous functions over noncommutative spaces, the local symmetries are reflected in some Hopf algebras. One of the first noncommutative spaces studied in NCG is the C ∗ -algebra of a foliated space. In the case of codimension n foliations, Connes and Moscovici discovered a Hopf algebra Hn, which governs the local symmetry of leaf spaces of foliations of codimension n. The Hopf algebra Hn is universal in the sense that it depends only on the codimension of a foliation. This family of Hopf algebras {Hn} is very useful in the study of index theory, and later was found to have connections with various different areas of mathematics, c.f. (10), (13). In this paper, we review the application of the Hopf algebra H1 in Rankin-Cohen deformations, which was initiated by Connes and Moscovici (14). We start by recalling the general setting of geometry. Let M be a smooth manifold and F be a foliation on M of codimension n. Let X be a complete flat transversal of F, and F + X be the oriented frame bundle of X. The holonomy pseudogroup acts on X and therefore F + X by transforming X parallelly along paths in leaves of F. The transverse geometry is to study the transversal X along with the action by the holonomy pseudogroup . In what follows we focus on the case when n = 1, and define Connes-Moscovici's Hopf algebra H1. Now the complete transversal X is a flat 1-dim manifold; and the oriented frame bundle F + X is diffeomorphic to X × R + , and is a discrete holonomy pseudogroup acting on X as local diffeomorphisms. We introduce coordinates x on X and y on R + . The lifted action of on F + X is

In this article, we prove that a free divisor in a three dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. F.J. Calderon-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem which describes in all dimensions a special minimal system of generators for the module of formal logarithmic vector fields. This formal structure theorem is closely related to the formal decomposition of a vector field by Kyoji Saito and is used in the proof of the above result. Another consequence of the formal structure theorem is that the truncated Lie algebras of logarithmic vector fields up to dimension three are solvable. We give an example that this may fail in higher dimensions.

A Baxter algebra is a commutative algebra A that carries a generalized integral operator. In the first part of this paper we review past work of Baxter, Miller, Rota and Cartier in this area and explain more recent work on explicit constructions of free Baxter algebras that extended the constructions of Rota and Cartier. In the second part of the paper we will use these explicit constructions to relate Baxter algebras to Hopf algebras and give applications of Baxter algebras to the umbral calculus in combinatorics.

We study a Hopf algebroid, \mathcal H , naturally associated to the groupoid U^δ_n ⋉ U_n . We show that classes in the Hopf cyclic cohomology of \mathcal H can be used to define secondary characteristic classes of trivialized flat U_n -bundles. For example, there is a cyclic class which corresponds to the universal transgressed Chern character and which gives rise to the continuous part of the ρ -invariant of Atiyah–Patodi–Singer. Moreover, these cyclic classes are shown to extend to pair with the K-theory of the associated C ^* -algebra. This point of view gives leads to homotopy invariance results for certain characteristic numbers. In particular, we define a subgroup of the cohomology of a group analogous to the Gelfand–Fuchs classes described by Connes [3] and show that the higher signatures associated to them are homotopy invariant.

Constructing Lie algebras of first order differential operators

We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebra \(\mathcal{H}_n\). More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of \(\mathcal{H}_n\), and we show that the van Est type characteristic homomorphism from the Hopf-cyclic complex of \(\mathcal{H}_n\) to the Gelfand–Fuks cohomology of the Lie algebra \(W_n\) of formal vector fields on \({\mathbb {R}}^n\) respects this multiplicative structure. We then illustrate the machinery for \(n=1\).

We study certain properties of the group of substitutions of formal power series in one variable with integer coefficients. We show that , regarded as a topological group, has four generators and cannot be generated by fewer elements. In particular, we show that the one-dimensional continuous homology of is isomorphic to . We study various topological and geometric properties of the coset space . We compute the real cohomology with uniformly locally constant supports and show that it is naturally isomorphic to the cohomology of the nilpotent part of the Lie algebra of formal vector fields on the line.

On the cohomology of the Lie algebra of formal vector fields preserving a flag