Automorphismes des algèbres cubiques

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Note aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. The Artin–Schelter classification of regular algebras of global dimension three contains two types of algebra: quadratic and cubic. Ewen and Ogievetsky classified the quantum matrix groups which are deformations of GL(3) corresponding to the quadratic algebras in the Artin–Schelter classification. In this Note we consider the cubic Artin–Schelter algebras as quantum spaces and construct Hopf algebras of their automorphisms. To cite this article: T. Popov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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.     

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH^?(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.     

Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1

We develop intrinsic tools for computing the periodic Hopf cyclic cohomology of Hopf algebras related to transverse symmetry in codimension 1. Besides the Hopf algebra found by Connes and the first author in their work on the local index formula for transversely hypoelliptic operators on foliations, this family includes its ‘Schwarzian’ quotient, on which the Rankin-Cohen universal deformation formula is based, the extended Connes-Kreimer Hopf algebra related to renormalization of divergences in QFT, as well as a series of cyclic coverings of these Hopf algebras, motivated by the treatment of transverse symmetry for non-orientable foliations.The method for calculating their Hopf cyclic cohomology is based on two computational devices, which work in tandem and complement each other: one is a spectral sequence for bicrossed product Hopf algebras and the other a Cartan-type homotopy formula in Hopf cyclic cohomology.     

Cohomological Obstructions to Cleft Extensions over Cocommutative Hopf Algebras

We extend the cohomological treatment of cleft extensions over cocommutative Hopf algebras by giving an interpretation of degree three Sweedler cohomology classes as obstructions to extensions. We show that every twisted action of a cocommutative Hopf algebra on an algebra R gives rise to an obstruction, a degree three Sweedler cohomology class of H with values in the center of R. The obstruction vanishes if and only if the twisted action belongs to a crossed product extension. We also show that every Sweedler three-cocycle can be realized as an obstruction.     

On the cohomology of highly connected covers of finite Hopf spaces

Relying on the computation of the André-Quillen homology groups for unstable Hopf algebras, we prove that if the mod p cohomology of both the fiber and the base in an H-fibration is finitely generated as algebra over the Steenrod algebra, then so is the mod p cohomology of the total space. In particular, the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra.     

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.     

Sur la cohomologie des algèbres de type AnOn the cohomology of type An algebras

We are interested here in the Hochschild cohomology of tensor triangular algebras $\text{T}$. We describe in particular a spectral sequence, whose terms are parametrized by the lengths of the trajectories of the quiver associated with $\text{T}$, and which converges to $\text{HH}{}^1\text{(}\text{T}\text{)}$, the Hochschild cohomology of $\text{T}$. Differentials at the first level are sums of cup products. To cite this article: S. Dourlens, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 527–532.     

Reflexive algebras with finite width lattices: Tensor products,cohomology, compact perturbations

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ¹-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg($\text{L}$₁ ? $\text{L}$₂) = Alg $\text{L}$₁ ? Alg $\text{L}$₂, where each $\text{L}$_i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each $\text{L}$_i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if $\text{L}$ is a finite-width commutative subspace lattice and $\text{K}$ is the set of compact operators then the quasitriangular algebra Alg $\text{L}$ + $\text{K}$ is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.     

Every finitely generated group is weakly exact

We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf G-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf G-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on ?_∞-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.     

On the Hopf–Schur group of a field

Let k be any field. We consider the Hopf–Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf–Schur group can be much larger than the Schur group of k.     

The Hochschild cohomology of square-free monomial complete intersections

We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular we provide an explicit formula for the cup product which gives the cohomology module an algebra structure and then we describe this structure in terms of generators and relations. In addition, we compute the Hilbert series of the Hochschild cohomology of these algebras.     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

From racks to pointed Hopf algebras

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces , where X is a rack and q is a 2-cocycle on X with values in . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.     

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.     

Homologies d'algèbres Artin–Schelter régulières cubiques

Hochschild homology of cubic Artin–Schelter regular algebras of type A with generic coefficients is computed. We follow the method used by Van den Bergh (K-Theory 8 (1994) 213–230) in the quadratic case, by considering these algebras as deformations of a polynomial algebra, with remarkable Poisson brackets. A new quasi-isomorphism is introduced. De Rham cohomology, cyclic and periodic cyclic homologies, and Hochschild cohomology are also computed. To cite this article: N. Marconnet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Cohomology theories of Hopf bimodules and cup-product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by Gerstenhaber and Schack, and by Ospel. We prove, when all the spaces involved are finite dimensional, that they are all equal to the Ext functor on the module category of an associative algebra X associated to A, as described by Cibils and Rosso. We also give an expression for a cup-product in the cohomology defined by Ospel, and prove that it corresponds to the Yoneda product of extensions.     

Hochschild Cohomology Rings of Temperley-Lieb Algebras

The authors first construct an explicit minimal projective bimodule resolution(P, δ) of the Temperley-Lieb algebra A, and then apply it to calculate the Hochschild cohomology groups and the cup product of the Hochschild cohomology ring of A based on a comultiplicative map Δ:P → PAP. As a consequence, the authors determine the multiplicative structure of Hochschild cohomology rings of both Temperley-Lieb algebras and representation-finite q-Schur algebras under the cup product by giving an explicit presentation by generators and relations.     

Lazy cohomology: An analogue of the Schur multiplier for arbitrary Hopf algebras

We propose a detailed systematic study of a group associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac-Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally, the explicit computation of for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.