Examples of support varieties for Hopf algebras with noncommutative tensor products

The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain a family of examples of such Hopf algebras and their modules, and classify left, right, and two-sided ideals in their stable module categories.     

Classification of pointed rank one Hopf algebras

We classify pointed rank one Hopf algebras over fields of prime characteristic which are generated as algebras by the first term of the coradical filtration. We obtain three types of Hopf algebras presented by generators and relations. For Hopf algebras with semi-simple coradical only the first and second type appears. We determine the indecomposable projective modules for certain classes of pointed rank one Hopf algebras.     

Quasi-triangular Hopf algebras and invariant Jacobians

We show that two module homomorphisms for groups and Lie algebras established by Xi can be generalized to the setting of quasi-triangular Hopf algebras. These module homomorphisms played a key role in his proof of a conjecture of Yau (1998). They will also be useful in the problem of decomposition of tensor products of modules. Additionally, we give another generalization of result of Xi in terms of Chevalley-Eilenberg complex.     

On corepresentations of multiplier Hopf algebras

We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.     

Yang-Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki-Koike-Shoji algebras

After reformulating the representation theory of 0-Hecke algebras in an appropriate family of Yang-Baxter bases, we investigate certain specializations of the Ariki-Koike algebras, obtained by setting q=0 in a suitably normalized version of Shoji's presentation. We classify the simple and projective modules, and describe restrictions, induction products, Cartan invariants and decomposition matrices. This allows us to identify the Grothendieck rings of the towers of algebras in terms of certain graded Hopf algebras known as the Mantaci-Reutenauer descent algebras, and Poirier quasi-symmetric functions. We also describe the Ext-quivers, and conclude with numerical tables.     

Properties of semisimple Hopf algebras

Semisimple finite-dimensional Hopf algebras all of whose non-one-dimensional irreducible modules of the same dimension are isomorphic are considered. The one-dimensional submodules of the tensor squares of irreduciblemodules are indicated and the almost cocommutative Hopf algebras and the left ideal coideals are described.     

Making the Category of Doi-Hopf Modules into a Braided Monoidal Category

We investigate how the category of Doi-Hopf modules can be made into a monoidal category. It suffices that the algebra and coalgebra in question are both bialgebras with some extra compatibility relation. We study tensor identies for monoidal categories of Doi-Hopf modules. Finally, we construct braidings on a monoidal category of Doi-Hopf modules. Our construction unifies quasitriangular and coquasitriangular Hopf algebras, and Yetter-Drinfel'd modules.     

Projective Modules Over Frobenius Algebras and Hopf Comodule Algebras

This note presents some results on projective modules and the Grothendieck groups K ₀ and G ₀ for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions.     

Indecomposable decomposition of tensor products of modules over Drinfeld doubles of Taft algebras

In this paper, we study the tensor product structure of the category of finite dimensional modules over Drinfeld doubles of Taft Hopf algebras. Tensor product decomposition rules for all finite dimensional indecomposable modules are explicitly given.     

Generalized Lie Algebras and Cocycle Twists

We use the results of Etingof and Gelaki on the classification of (co)triangular Hopf algebras to extend Scheunert's “discoloration” technique to Lie algebras in the category of (co)modules. As an application, we prove a PBW-type theorem for such Lie algebras. We also discuss the relationship between Lie algebras in the category of (co)modules and symmetric braided Lie algebras introduced by Gurevich. Finally, we construct examples of symmetric braided Lie algebras that are essentially different from Lie coloralgebras.     

Module twistors for modules of nonlocal vertex algebras

We introduce the notion of module twistor for a module of a nonlocal vertex algebra. The aim of this paper is to use this concept to unify some deformed constructions of modules of nonlocal vertex algebras, such as twisted tensor products and iterated twisted tensor products of modules of nonlocal vertex algebras.     

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.     

Nichols algebras over weak Hopf algebras

In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra H.Although the category of all left H-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra H, and a series of weak Hopf algebras. Some results of [8] are generalized.     

Determining all indecomposable codes over some Hopf algebras

We determine all indecomposable codes over a class of Hopf algebras named Taft Algebras. We calculate dual codes and tensor products of these indecomposable codes and give applications of them.     

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.     

On the order of the antipode of hopf algebras

In this paper we consider a conjecture on the order of the antipode of semisimple Hopf algebras in the Yetter-Drinfeld category and study a related property of the ordinary Hopf algebras. We show that most known examples of finite-dimensional semisimple Hopf algebras satisfy this property.     

Hopf Categories

We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.     

Algebras, hyperalgebras, nonassociative bialgebras and loops

Sabinin algebras are a broad generalization of Lie algebras that include Lie, Malcev and Bol algebras as very particular examples. We present a construction of a universal enveloping algebra for Sabinin algebras, and the corresponding Poincaré-Birkhoff-Witt Theorem. A nonassociative counterpart of Hopf algebras is also introduced and a version of the Milnor-Moore Theorem is proved. Loop algebras and universal enveloping algebras of Sabinin algebras are natural examples of these nonassociative Hopf algebras. Identities of loops move to identities of nonassociative Hopf algebras by a linearizing process. In this way, nonassociative algebras and Hopf algebras interlace smoothly.