Projective Hopf Algebra Research Articles

Duality for Finite Hopf Algebras Explained by Corings

We give a coring version for the duality theorem for actions and coactions of a finitely generated projective Hopf algebra. We also provide a coring analogue for a theorem of H.-J. Schneider, which generalizes and unifies the duality theorem for finite Hopf algebras and its refinements.

Integrals in Hopf Algebras over Rings#‡

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.

FROBENIUS FUNCTORS OF THE SECOND KIND

ABSTRACT A pair of adjoint functors is called a Frobenius pair of the second type if is a left adjoint of for some category equivalences and . Frobenius ring extensions of the second kind provide examples of Frobenius pairs of the second kind. We study Frobenius pairs of the second kind between categories of modules, comodules, and comodules over a coring. We recover the result that a finitely generated projective Hopf algebra over a commutative ring is always a Frobenius extension of the second kind (cf.[27], [17], and prove that the integral spaces of the Hopf algebra and its dual are isomorphic.

An approach to Hopf algebras via Frobenius coordinates II

We make a study of finitely generated, projective Hopf algebras over commutative rings from the point of view of P-Frobenius algebras. Defining modular function as the composite of counit with Nakayama automorphism, we establish the Radford formula (Amer. J. Math. 98 (1976) 333–335) [34] by means of comparing and transforming a Frobenius system. We then study when such Hopf algebras are separable and strongly separable, their Hopf subalgebras as Frobenius extensions of a third kind, their Drinfeld double and a change to a ground ring with trivial Picard group.

Morphisms of Relative Hopf Modules, Smash Products, and Duality

The aim of this paper is to extend a result of H.-J. Schneider on endomorphism rings of relative Hopf modules [17, Theorem 3.2] to the case of projective Hopf algebras with bijective antipode, and to show the connection between this extension and several duality theorems for Hopf algebras. A consequence of this generalization is the following: if H is a co-Frobenius Hopf algebra over the field k, A/B is a Hopf-Galois extension and M is a right (A,H)-Hopf module, then, considering the usual ring embeddings

Embedding the Hopf Automorphism Group into the Brauer Group

AbstractLet H be a faithfully projective Hopf algebra over a commutative ring k. In [8, 9] we defined the Brauer group BQ(k, H) of H and an homomorphism π from Hopf automorphism group AutHopf(H) to BQ(k,H). In this paper, we show that the morphism π can be embedded into an exact sequence.

Computing the Brauer-Long group of a Hopf algebra II: the Skolem-Noether theory

Let H be a commutative, cocommutative and faithfully projective Hopf algebra over a commutative ring R. Then Long's Brauer group of H-dimodule algebras fits into an exact sequence 1→BD s(R,H)→BD(R,H) → β O(R,H) min, where O(R,H) min is a well-defined subgroup of the group of Hopf algebra automorphisms of H⊗ H ∗. It is not known whether β is surjective, but some partial results are given. The theory is applied to Orzech's and Deegan's subgroups of the Brauer-Long group.

A note on inner actions of Hopf algebras

Let H H be a commutative, cocommutative, and faithfully projective Hopf algebra over a commutative ring R R . A twisted version of inner action of a Hopf algebra, called H H -inner action is introduced, and it is shown that H H acts H H -innerly on an H H -Azumaya algebra, if Pic ⁡ ( H ∗ ) \operatorname {Pic} ({H^ * }) is trivial.

Computing the brauer-long group of a hopf algebra I: The cohomological theory

LetH be a commutative, cocommutative and faithfully projective Hopf algebra over a commutative ringR. Using cohomological methods, we obtain a description of a subgroup of Long’s Brauer group ofH-dimodule algebras:Ψ where the mapψ is induced by the smash product, and where BDs (R, H) is the subgroup of the Brauer-Long group consisting of all elements which are split by a faithfully flat extension ofR. As an example, the Brauer-Long group of a free Hopf algebra of rank 2 is computed. The results are also applied to Orzech’s subgroup of the Brauer-Long group.

Hopf algebras over number rings

Sweedler [7, p. 891 first mentioned that the antipodes of Hopf algebras over fields may have any even order or infinite order. The examples he gave are infinite dimensional. Radford [S] proved later that the order of the antipode of a finite dimensional Hopf algebra over a field must be finite. A sequence of papers published by Taft, Wilson, and Radford [6,9, lo], showed that it is possible to construct finite dimensional Hopf algebras over any field with antipodes of any given even order. Waterhouse [ 111 generalized the finiteness of antipodal order to finitely generated projective Hopf algebras over any commutative ring. So, there is a natural question of whether we could construct Hopf algebras whose antipodal orders are arbitrary even integers at the level of commutative rings. In this paper we give positive answers for number rings Z [ l/n]. Originally, Taft [S] introduced examples of finite dimensional Hopf algebras with antipodes of arbitrary even order 2n over any field that contains a primitive nth root of unity. For fields of characteristic 0 or p not dividing n this restriction was dropped later in a joint work by him, Radford, and Wilson [6]. (Note: for general fields, see [lo].) The main ideas there are using a finite Galois extension of the base field to recover all primitive nth roots, constructing one copy of the Hopf algebra for each root, tensoring these together, defining a group of semilinear automorphisms with respect to the Galois group, taking the set of fixed elements, and obtaining the required Hopf algebra as a form. These two results are restated in Section 2. In Section 2 we present basic notations and conventions used in later discussion. Some elementary results following from these notions are quickly pointed out.

Galois Objects of Finitely Generated Projective Hopf Algebras

Let C be the category of cocommutative coalgebras over a commutative ring R and let H be a group object in C, i.e., let H be a cocommutative Hopf algebra. Assume that H is a finitely generated, projective R-module and that the integrals (of [4]) in H* ≡ HomR(H, R) are cocommutative elements. We will show that any Galois H-object (as defined in [3, Def. 1.2, p. 8]) is a finitely generated, projective R-module.

On the cohomology of modules over Hopf algebras

Let R be a commutative ring. Define an FH-algebra H to be a Hopf algebra and a Frobenius algebra over R with a Frobenius homomorphism ψ such that ∑ ( h) h (1) ψ( h (2)) = ψ( h) · 1 for all h ϵ H. This is essentially the same as to consider finitely generated projective Hopf algebras with antipode. For modules over FH-algebras we develop a cohomology theory which is a generalization of the cohomology of finite groups. It generalizes also the cohomology of finite-dimensional restricted Lie algebras. In particular the following results are shown. The complete homology can be described in terms of the complete cohomology. There is a cup-product for the complete cohomology and some of the theorems for periodic cohomology of finite groups can be generalized. We also prove a duality theorem which expresses the cohomology of the “dual” of an H-module as the “dual” of the cohomology of the module. The last section provides techniques to describe under certain conditions the cohomology of H by the cohomology of sub- and quotient-algebras of H. In particular we have a generalization of the Hochschild-Serre spectral sequence for the cohomology of groups.

When Hopf algebras are Frobenius algebras

R. Larson and M. Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper I give a generalization of this result which needs only a minor restriction, which, for example, always holds if pic( R) = 0 for the base ring R. A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ψ such that Σ h (1) ψ( h (2)) = ψ( h) · 1 for all h ϵ H. We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, R-direct summand of H.