Hopf Algebroid Research Articles

REPRESENTATIONS OF INTERNAL CATEGORIES

Adams gave the notion of a Hopf algebroid generalizing the notion of a Hopf algebra and showed that certain generalized homology theories take values in the category of comodules over the Hopf algebroid associated with each homology theory. A Hopf algebra represents an affine group scheme which is a group in the category of a scheme and the notion of comodules over a Hopf algebra is equivalent to the notion of representations of the affine group scheme represented by a Hopf algebra. On the other hand, a Hopf algebroid represents a groupoid in the category of schemes. Therefore, it is natural to consider the notion of comodules over a Hopf algebroid as representations of the groupoid represented by a Hopf algebroid. This motivates the study of representations of groupoids, and more generally categories, for topologists. The aim of this paper is to set a categorical foundation of representations of an internal category which is a category object in a given category, using the notion of a fibered category.

On affine morphisms of Hopf algebroids

The paper considers the FPQC stacks which are associated to affine groupoid schemes. Using a formulation of a descent datum in terms of morphisms of affine groupoid schemes, explicit arguments are given which avoid appeal to the general principle of faithfully flat descent. This theory is applied to consider the notion of affine morphism.

The ring spectrum $P(n)$ for the prime 2

This paper is a companion to the Boardman‐Wilson paper on the ring spectrum P(n). When the prime is 2, this spectrum is not commutative, which introduces several complications. Here, we supply the necessary details of the relevant Hopf algebroids and Hopf ring for this case.

A Schneider type theorem for Hopf algebroids

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ A by H , are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (non-commutative) base algebra of H , relative injectivity of the H -comodule algebra A is related to the Galois property of the extension B ⊆ A and also to the equivalence of the category of relative Hopf modules to the category of B-modules. This extends a classical theorem by H.-J. Schneider on Galois extensions by a Hopf algebra. Our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor, a notion introduced and analysed hereby.

Anchor Maps and Stable Modules in Depth Two

An algebra extension A ∣ B is right depth two if its tensor-square \(A \otimes _{B} A\) is in the Dress category \({\mathbf{Add}}_{A} A_{B}\). We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids \(S = {\text{End}}_{B} A_{B}\) and \(T = {\text{End}}_{A} A \otimes _{B} A_{A}\) over the centralizer R = C A (B) are the modules S R and R T studied in Kadison (J. Alg. & Appl., 2005, preprint), Kadison (Contemp. Math., 391: 149–156, 2005), and Kadison and Külshammer (Commun. Algebra, 34: 3103–3122, 2006), which provide information about the bialgebroids and the extension (Kadison, Bull. Belg. Math. Soc. Simon Stevin, 12: 275–293, 2005). The anchor maps for the Hopf algebroids in Khalkhali and Rangipour (Lett. Math. Phys., 70: 259–272, 2004) and Kadison (2005, preprint) reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in Van Oystaeyen and Panaite (Appl. Categ. Struct., 2006, in press). We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in Kadison (Proc. Am. Math. Soc., 131: 2993–3002, 2003 Prop. 1.1) to any A-module. We observe that Schneider’s coGalois theory in Schneider (Isr. J. Math., 72: 167–195, 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.

Categorical Methods in Hopf Algebras

The main idea behind many notions in Ring Theory and other disciplines can be formulated in a natural way in a categorical setting. For example, a natural way to introduce the notion of bialgebra is the following: bialgebra structures on an algebra A correspond bijectively to monoidal structures on the category of representations of A, such that the forgetful functor to the category of vector spaces is strongly monoidal. If we require, in addition, that the category of representations has duality, then we recover the definition of a Hopf algebra. Other concepts, that appeared more recently in connection to quantum group theory, such as quasi-triangular Hopf algebras, quasi-Hopf algebras, weak Hopf algebras and Hopf algebroids, can be explained in a similar way, using (braided) monoidal categories. This special issue contains a collection of recent articles on Hopf algebras and related topics, reflecting this philosophy. An important recent trend is the revival of the theory of corings. Corings can be considered as coalgebras over non-commutative rings; they were introduced by Sweedler in 1975. A renewed interest in the topic began in 2000, with a paper by Brzezinski, in which it was explained how a series of results on Hopf modules and related topics can be unified and reformulated more elegantly in terms of corings. Six contributions in this issue are directly connected to corings. The most categorical of them is the paper by Gomez–Torrecillas, explaining connections between Galois theory for corings and comonads. Three other papers (by Bohm and Brzezinski, by Kadison, and by Panaite and Van Oystaeyen) are about Hopf algebroids. Hopf algebroids can be viewed as Hopf algebras over non-commutative rings, but, as indicated above, they can be interpreted in terms of monoidal categories.

Cleft Extensions of Hopf Algebroids

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid ℋ = (ℋ L , ℋ R )) is cleft if and only if it is ℋ R -Galois and has a normal basis property relative to the base ring L of ℋ L . Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the Chern–Galois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined.

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M , the groupoid convolution algebra C c ∞ ( G ) of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra C c ∞ ( M ) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C c ∞ ( M ) we construct the associated spectral étale Lie groupoid G sp ( A ) over M such that G sp ( C c ∞ ( G ) ) is naturally isomorphic to G . Both these constructions are functorial, and C c ∞ is fully faithful left adjoint to G sp . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid C c ∞ ( G ) of an étale Lie groupoid G .

Products of Linear Maps on Bialgebras, with Applications to Left Hopf Algebras and Hopf Algebroids

The Doi–Koppinen generalized smash product can be applied to linear maps on bialgebras, and we define a new associative product for linear maps on bialgebras. We comment on the applications of these products to left Hopf algebras and Hopf algebroids.

Realizability of the Adams-Novikov spectral sequence for formal $A$-modules

We show that the formal A-module Adams-Novikov spectral sequence of Ravenel does not naturally arise from a filtration on a map of spectra by examining the case A = Z[i]. We also prove that when A is the ring of integers in a nontrivial extension of Qp, the map (L, W) → (L A , W A ) of Hopf algebroids, classifying formal groups and formal A-modules respectively, does not arise from compatible maps of E ∞ -ring spectra (MU, MU^MU)→ (R, S).

Morita theory for coring extensions and cleft bicomodules

A Morita context is constructed for any comodule of a coring and, more generally, for an L - C bicomodule Σ for a coring extension ( D : L ) of ( C : A ) . It is related to a 2-object subcategory of the category of k-linear functors M C → M D . Strictness of the Morita context is shown to imply the Galois property of Σ as a C -comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an L - C bicomodule Σ—implying strictness of the associated Morita context—is introduced. It is shown to be equivalent to being a Galois C -comodule and isomorphic to End C ( Σ ) ⊗ L D , in the category of left modules for the ring End C ( Σ ) and right comodules for the coring D , i.e. satisfying the normal basis property. Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules.

Finitary Galois extensions over noncommutative bases

We study Galois extensions M ( co - ) H ⊂ M for H-(co)module algebras M if H is a Frobenius Hopf algebroid. The relation between the action and coaction pictures is analogous to that found in Hopf–Galois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various classical theorems of Kreimer–Takeuchi, Doi–Takeuchi and Cohen–Fischman–Montgomery. We find that the Galois extensions N ⊂ M over some Frobenius Hopf algebroid are precisely the balanced depth 2 Frobenius extensions. We prove that the Yetter–Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by a slightly generalized Brzeziński–Militaru theorem. Contravariant “fiber functors” are used to prove an analogue of Ulbrich's theorem and to get a monoidal embedding of the module category M E of the endomorphism Hopf algebroid E = End M N N into M N op N .

Integral Theory for Hopf Algebroids

The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.

Morita contexts on the Hopf algebroid

Morita contexts on the Hopf algebroid

Hopf algebroids and Galois extensions

To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := End\,_BA_B$ and $T := (A \o_B A)^B$ over the centralizer $R$ using the depth two theory in [18,Kadison-Szlachányi]. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra [21,8] to depth two extensions using coring theory [3]. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem \cite[23,5.1] for smash products with special modules over the Drinfel'd double, the Miyashita-Ulbrich action, the fact that $R$ is a commutative algebra in the pre-braided category of Yetter-Drinfel'd modules [28] and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double [24]. In our last section, an exposition of results of Sugano [29,30] leads us to a Galois correspondence between sub-Hopf algebroids of $S$ over simple subalgebras of the centralizer with finite projective intermediate simple subrings of a finite projective H-separable extension of simple rings $A \supseteq B$.

Recognizing Hopf algebroids defined by a group action

Let A be a complete noetherian regular local ring, and suppose that S is a profinite group acting continuously on A via ring homomorphisms. Let Γ = Map c ( S , A ) , the algebra of continuous functions from S to A. Then ( A , Γ ) has a canonical structure of a complete Hopf algebroid, determined by the action of S on A. We give necessary and sufficient conditions for a general complete Hopf algebroid to be of this form. Applications to Morava theory are also discussed.

Morita contexts on the Hopf algebroid

We generalize the Morita theory to the Hopf algebroid and give the characterization of the surjectivity of one of the Morita context maps.

Galois theory for Hopf algebroids

An extensionB⊂A of algebras over a commutative ringk is anH-extension for anL-bialgebroidH ifA is anH-comodule algebra andB is the subalgebra of its coinvariants. It isH-Galois if the canonical mapA ⊗BA →A ⊗LH is an isomorphism or, equivalently, if the canonical coringA ⊗LH:A is a Galois coring.

Hopf Algebroid Symmetry of Abstract Frobenius Extensions of Depth 2#

We study Frobenius 1-cells ι in an additive bicategory 𝒞 satisfying the depth 2 condition. We show that the rings of 2-cells 𝒞2 and can be equipped with dual Hopf algebroid structures. We prove also that a Hopf algebroid appears as the solution of the above abstract symmetry problem if and only if it possesses a two sided non-degenerate integral.

Para-Hopf Algebroids and their Cyclic Cohomology

We introduce the concept of para-Hopf algebroid and define their cyclic cohomology in the spirit of Connes–Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, including the Connes–Moscovici algebra \({\cal H}_{FM}\) , are para-Hopf algebroids