A proof that commutative artinian rings are noetherian

Abstract

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.     

Maschke’s theorem for smash products of quasitriangular weak Hopf algebras

The paper is concerned with the semisimplicity of smash products of quasitriangular weak Hopf algebras. Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k and A any semisimple and quantum commutative weak H-module algebra. Based on the work of Nikshych et al. (Topol. Appl. 127(1–2):91–123, 2003), we give Maschke’s theorem for smash products of quasitriangular weak Hopf algebras, stating that A#H is semisimple if and only if A is a projective left A#H-module, which extends the Theorem 3.2 given in Yang and Wang (Commun. Algebra 27(3):1165–1170, 1999).     

Twisted Smash Products and L-R Smash Products for Biquasimodule Hopf Quasigroups

We consider two new algebras from an H-biquasimodule algebra A and a Hopf quasigroup H: twisted smash product A ? H and L-R smash product A?H, and find necessary and sufficient conditions for making them Hopf quasigroups. We generalize the main results in Brzeziński and Jiao [5] and Klim and Majid [9]. Moreover, if H is a cocommutative Hopf quasigroup, we prove that A ? H is isomorphic to A?H as Hopf quasigroups.     

An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.     

Quasitriangular Pointed Hopf Algebras Constructed by Ore Extensions

We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft's Hopf algebras H _n ², Radford's Hopf algebras H _N,n,q, and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel'd double.     

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ?-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ?-situation.     

Representation theory of Hopf galois extensions

LetH be a Hopf algebra over the fieldk andB ⊂A a right faithfully flat rightH-Galois extension. The aim of this paper is to study some questions of representation theory connected with the ring extensionB ⊂A, such as induction and restriction of simple or indecomposable modules. In particular, generalizations are given of classical results of Clifford, Green and Blattner on representations of groups and Lie algebras. The stabilizer of a leftB-module is introduced as a subcoalgebra ofH. Very often the stabilizer is a Hopf subalgebra. The special case whenA is a finite dimensional cocommutative Hopf algebra over an algebraically closed field,B is a normal Hopf subalgebra andH is the quotient Hopf algebra was studied before by Voigt using the language of finite group schemes.     

Basic quasi-Hopf algebras over cyclic groups

Let m be a positive integer, not divisible by 2, 3, 5, 7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in Etingof and Gelaki (2006) [11] to the case of cyclic groups of order m. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras A(H,s), constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order m is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type A(H,s).     

On Smash Products Of Hopf Algebras

Let H = X? _R A denote an R-smash product of the two bialgebras X and A. We prove that (X,A) is a pair of matched bialgebras, if the R-smash product H has a braiding structure. When X is an associative algebra and A is a Hopf algebra, we investigate the global dimension and the weak dimension of the smash product H = X? _R A and show that lD(H) ≤ rD(A) + lD(X) and wD(H) ≤ wD(A) + wD(X). As an application, we get lD(H ₄) = ∞ for Sweedler's four dimensional Hopf algebra H ₄. We also study the associativity of smash products and the relations between smash products and factorization for algebras.     

Faithfully Flat Hopf Bi-Galois Extensions

This article is devoted to faithfully flat Hopf bi-Galois extensions defined by Fischman, Montgomery, and Schneider. Let H be a Hopf algebra with bijective antipode. Given a faithfully flat right H-Galois extension A/R and a right H-comodule subalgebra C ? A such that A is faithfully flat over C, we provide necessary and sufficient conditions for the existence of a Hopf algebra W so that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra. As a consequence, we prove that if R = k, there is a Hopf algebra W such that A/C is a left W-Galois extension and A a (W, H)-bicomodule algebra if and only if C is an H-submodule of A with respect to the Miyashita–Ulbrich action.     

Actions of Ore extensions and growth of polynomial H-identities

We show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexp^H(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.     

Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$algebras

Let G be a group and assume that (A _p ) _p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δ _p,q : A _pq → A _p ⊗ A _q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ _p,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras. Presented by Ken Goodearl.     

A Lot of Quasitriangular Group-cograded Multiplier Hopf Algebras

A new method of constructing quasitriangular group-cograded multiplier Hopf algebras is provided. For a multiplier Hopf dual pairing σ between regular multiplier Hopf algebras A and B, we introduce the concept of a σ-compatible pairing (Φ, Ψ, σ ), where Φ and Ψ are actions of the twisted semi-direct group of a group G on A and B, respectively. We construct a twisted double group-cograded multiplier Hopf algbera D(A, B; σ, Φ, Ψ). Furthermore, if there is a canonical multiplier in M(B ⊗ A) we show existence of quasitriangular structure on D(A, B; σ, Φ, Ψ). As an application, special cases and examples are given.     

De-Equivariantization of Hopf Algebras

We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of coquasi-Hopf algebras A(H, G, Φ) attached to a coradically-graded pointed Hopf algebra H and some extra data.     

The wedderburn-malcev theorem for comodule algebras

Let H be a Hopf algebra over a field k. Under some assumptions on H we state and prove a generalization of the Wedderburn-Malcev theorem for i7-comodule algebras. We show that our version of this theorem holds for a large enough class of Hopf algebras, such as coordinate rings of completely reducible affine algebraic groups, finite dimensional Hopf algebras over fields of characteristic 0 and group algebras. Some dual results are also included.     

Co-Frobenius Hopf algebras and the coradical filtration

We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. Let H be a Hopf algebra whose coradical is a Hopf algebra. Let gr H be the associated graded coalgebra and let R be the diagram of H, c. f. [2]. Then the following are equivalent: (1) H is co-Frobenius; (2) gr H is co-Frobenius; (3) R is finite dimensional; (4) the coradical filtration of H is finite. This Theorem allows us to give systematically examples of co-Frobenius Hopf algebras, and opens the way to the classification of ample classes of such Hopf algebras. Received: 8 March 2001 / Published online: 8 November 2002 RID="*" ID="*" Parts of this work were done during visits of the second author to the University of Córdoba, Argentina, in May 2000; and of the first author to the University of Bucharest in November 2000. We thank the FOMEC and CNCSIS (Grant C12) for support to these visits. The first author also thanks ANPCyT, CONICET, Agencia Córdoba Ciencia and Secyt (UNC) for partial support. The second author was partiall y supported by Grant SM10/01 of the Research Administration of Kuwait University.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

Basic Hopf algebras and quantum groups

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers given in terms of a finite group G and sequence of elements of G in the following way can occur. The quiver has vertices and arrows , where the set is closed under conjugation with elements in G and for each g in G, the sequences W and are the same up to a permutation. We show how is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R]. Received August 15, 1994; in final form May 16, 1997     

Breuil–Kisin Modules and Hopf Orders in Cyclic Group Rings

For K, a finite extension of ? _p with ring of integers R, we show how Breuil–Kisin modules can be used to determine Hopf orders in K-Hopf algebras of p-power dimension. We find all cyclic Breuil–Kisin modules and use them to compute all of the Hopf orders in the group ring KΓ where Γ is cyclic of order p or p ². We also give a Laurent series interpretation of the Breuil–Kisin modules that give these Hopf orders.