Juan Cuadra

IDEMPOTENTS AND MORITA-TAKEUCHI THEORY

ABSTRACT We offer an approach to basic coalgebras with inspiration in the classical theory of idempotents for finite dimensional algebras. Our theory is based upon the fact that the co-hom functors associated to direct summands of the coalgebra can be easily described in terms of idempotents of the convolution algebra. Our approach is shown to be equivalent to that given by W. Chin and S. Montgomery by using co-endomorphism coalgebras of minimal injective cogenerators.

On the structure of (co-Frobenius) Hopf algebras

We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).

Graded almost noetherian rings and applications to coalgebras

It is well-known that the dual algebra of a coalgebra C is a topological algebra with the weak-∗ topology. In this paper we study some finiteness conditions relative to the topological structure of C∗ in terms of the category Rat(C∗M) of rational left C∗-modules. In particular, we focus on the problem whether Rat(C∗M) is closed under extensions. In torsion theoretic terms this raises the question of deciding when Rat(C∗M) is a torsion theory or a localizing subcategory in C∗M, the category of all left C∗-modules (the notion of localizing subcategory used here is as in [5], [19]). This problem has been previously treated in [9], [11], and [18], where a coalgebra satisfying this property is said to be a coalgebra having a torsion rat functor.

Flat Comodules and Perfect Coalgebras

Stenström introduced the notion of flat object in a locally finitely presented Grothendieck category 𝒜. In this article we investigate this notion in the particular case of the category 𝒜 = C-Comod of left C-comodules, where C is a coalgebra over a field K. Several characterizations of flat left C-comodules are given and coalgebras having enough flat left C-comodules are studied. It is shown how far these coalgebras are from being left semiperfect. As a consequence, we give new characterizations of a left semiperfect coalgebra in terms of flat comodules. Left perfect coalgebras are introduced and characterized in analogy with Basss Theorem P. Coalgebras whose injective left C-comodules are flat are discussed and related to quasi-coFrobenius coalgebras.

The Brauer group of some quasitriangular Hopf algebras

Abstract We show that the Brauer group BM(k,Hν,Rs,β) of the quasitriangular Hopf algebra (Hν,Rs,β) is a direct product of the additive group of the field k and the classical Brauer group B θ s (k, Z 2ν ) associated to the bicharacter θs on Z 2ν defined by θs(x,y)=ωsxy, with ω a 2νth root of unity.

On Hopf Algebras with Nonzero Integral

Andruskiewitsch and Dăscălescu (2003) conjectured that any co-Frobenius Hopf algebra has finite coradical filtration. In this paper we provide two conditions under which this conjecture is true. One of these conditions is used to prove that the conjecture holds for the Hopf algebra of rational functions of an algebraic group with integral over a perfect field.

A Sequence to Compute the Brauer Group of Certain Quasi-Triangular Hopf Algebras

A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category

Subgroups of the Brauer Group of a Cocommutative Coalgebra

{{\mathcal C}}

Hopf Algebras and Tensor Categories

, and under certain assumptions on the braiding (fulfilled if

A Hopf algebra having a separable Galois extension is finite dimensional

{{\mathcal C}}