Hopf cyclic cohomology and transverse characteristic classes

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan–Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand–Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of étale foliation groupoids.     

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.     

Cyclic Homology of Hopf Algebras

A cyclic cohomology theory adapted to Hopf algebras has been introduced recently by Connes and Moscovici. In this paper, we consider this object in the homological framework, in the spirit of Loday and Quillen and Karoubi's work on the cyclic homology of associative algebras. In the case of group algebras, we interpret the decomposition of the classical cyclic homology of a group algebra in terms of this homology. We also compute both cyclic homologies for truncated quiver algebras.     

Cup products in Hopf-cyclic cohomology

We construct cup products of two different kinds for Hopf-cyclic cohomology. When the Hopf algebra reduces to the ground field our first cup product reduces to Connes' cup product in ordinary cyclic cohomology. The second cup product generalizes Connes–Moscovici's characteristic map for actions of Hopf algebras on algebras. To cite this article: M. Khalkhali, B. Rangipour, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.     

On the First Hochschild Cohomology of Admissible Algebras

The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras.For this purpose,the authors study differential operators on an admissible algebra.Firstly,differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed.Based on this discussion,the first cohomology with admissible algebras as coefficient modules is characterized,including their dimension formula.Besides,for planar quivers,the fc-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field fc of characteristic 0.     

On the First Hochschild Cohomology linebreak of Admissible Algebras

The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the $k$-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field $k$ of characteristic $0$.     

Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions.     

Cyclic cohomology of Hopf algebras

We give a construction of the Connes–Moscovici cyclic cohomology for any Hopf algebra equipped with a character (solving the technical problem raised in Connes and Moscovici (Comm. Math. Phys. 198 (1998) 199–246)). Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usual Chern–Weil theory.     

New coefficients for Hopf Cyclic Cohomology

In this note, the categories of coefficients for Hopf cyclic cohomology of comodule algebras and comodule coalgebras are extended. We show that these new categories have two proper different subcategories where the smallest one is the known category of stable anti Yetter–Drinfeld modules. We prove that components of Hopf cyclic cohomology such as cup products work well with these new coefficients.     

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.     

Quasi-Piecewise-Koszul Modules

In this article we first study an equivariant cyclic cohomology for weak H-module agebras over a weak Hopf algebra H with a bijective antipode. Then we define an equivariant K-theory for weak quantum Yetter–Drinfeld algebras over H and establish a generalized Connes' pairing between the equivariant cyclic cohomology and the equivariant K-theory. As an application we consider our theory for groupoids.     

Bivariant Hopf Cyclic Cohomology

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes' cyclic category Λ. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic cohomology of Connes and Moscovici and its dual version by fixing either one of the variables as the ground field. We also prove an appropriate version of Morita invariance for both of these theories.     

Invariant Cyclic Homology

We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple (A, H, M) consisting of a Hopf algebra H, an H-comodule algebra A, an H-module M, and a compatible grouplike element in H, we define the cyclic module of invariant chains on A with coefficients in M and call its cyclic homology the invariant cyclic homology of A with coefficients in M. We also develop a dual theory for coalgebras. Examples include cyclic cohomology of Hopf algebras defined by Connes–Moscovici and its dual theory. We establish various results and computations including one for the quantum group SL(q,2).     

Radford's S4 formula for co-Frobenius Hopf algebras

This note extends Radford's formula for the fourth power of the antipode of a finite-dimensional Hopf algebra to co-Frobenius Hopf algebras and studies equivalent conditions to a Hopf algebra being involutory for finite-dimensional and co-Frobenius Hopf algebras.     

Hopf Algebra Extensions of Group Algebras and Tambara-Yamagami Categories

We determine the structure of Hopf algebras that admit an extension of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension < 36 is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs.     

Skew Derivations and Deformations of a Family of Group Crossed Products

We obtain deformations of a crossed product of a polynomial algebra with a group, under some conditions, from universal deformation formulas. We show that the resulting deformations are nontrivial by a comparison with Hochschild cohomology. The universal deformation formulas arise from actions of Hopf algebras generated by automorphisms and skew derivations, and are universal in the sense that they apply to deform all algebras with such Hopf algebra actions.