If H is a quasi-Hopf algebra and B is a right H-comodule algebra such that there exists v:H\to B a morphism of right H-comodule algebras, we prove that there exists a left H-module algebra A such that B\simeq A# H. The main difference comparing to the Hopf case is that, from the multiplication of B, which is associative, we have to obtain the multiplication of A, which in general is not; for this we use a canonical projection E arising from the fact that B becomes a quasi-Hopf H-bimodule.
In this paper, we introduce the conception of Rota-Baxter paired comodules, which is dual to Rota-Baxter paired modules in [14]. We mainly discuss some properties of Rota-Baxter paired comodules, especially we give the characterization of generic Rota-Baxter paired comodules, which has important application for the construction of Rota-Baxter comodules. Moreover, we construct Rota-Baxter paired comodules on Hopf algebras, weak Hopf algebras, weak Hopf modules, dimodules, relative Hopf modules and Rota-Baxter paired comodules. And then we finally introduce the conception of Rota-Baxter paired H…
Let A be a commutative unital algebra over an algebraically closed field k of characteristic not equal to 2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let Q be a Hopf algebra that coacts on A inner-faithfully, while leaving V invariant. We prove that Q must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) Q is co-semisimple, finite-dimensional, and char(k)=0.
We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford's $S^4$ formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a monoidal isomorphism between certain double dual functors.
This is a sequel paper of arXiv:1306.1466 in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to …
After recalling the definitions and the properties of the generic base algebra and of the universal comodule algebra attached to a Hopf algebra by Aljadeff and the second-named author, we determine these algebras for the Taft algebras, the Hopf algebras E(n), and certain monomial Hopf algebras.
Principal comodule algebras can be thought of as objects representing principal bundles in non-commutative geometry. A crucial component of a principal comodule algebra is a strong connection map. For some applications it suffices to prove that such a map exists, but for others, such as computing the associated bundle projectors or Chern-Galois characters, an explicit formula for a strong connection is necessary. It has been known for some time how to construct a strong connection map on a multi-pullback comodule algebra from strong connections on multi-pullback components, but the known expli…
The aim of this paper is an algebraic study of the Hopf algebra H_R of rooted trees, which was introduced in \cite{Kreimer1,Connes,Broadhurst,Kreimer2}. We first construct comodules over H_R from finite families of primitive elements. Furthermore, we classify these comodules by restricting the possible families of primitive elements, and taking the quotient by the action of certain groups. In the next section, we give a formula about primitive elements of the subalgebra of ladders, and construct a projection operator on the space of primitive elements. It allows us to obtain a basis of the pri…
We construct the join of noncommutative Galois objects (quantum torsors) over a Hopf algebra H. To ensure that the join algebra enjoys the natural (diagonal) coaction of H, we braid the tensor product of the Galois objects. Then we show that this coaction is principal. Our examples are built from the noncommutative torus with the natural free action of the classical torus, and arbitrary anti-Drinfeld doubles of finite-dimensional Hopf algebras. The former yields a noncommutative deformation of a non-trivial torus bundle, and the latter a finite quantum covering.
This is a survey of results obtained jointly with E. Aljadeff and published in Adv. Math. 218 (2008), 1453-1495. We explain how to set up a theory of polynomial identities for comodule algebras over a Hopf algebra, and concentrate on the universal comodule algebra constructed from the identities satisfied by a given comodule algebra. All concepts are illustrated with various examples.
Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B\subseteq 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\subseteq 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…
An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with J. Ecalle's mould calculus, i.e. mould composition and contracting arborification, is precised.
We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces…
The duality between partial actions (partial $H$-module algebras) and co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully explored in this work. A connection between partial (co)actions and Hopf algebroids is established under certain commutativity conditions. Moreover, we continue this duality study, introducing also partial $H$-module coalgebras and their associated $C$-rings, partial $H$-comodule coalgebras and their associated cosmash coproducts, as well as the mutual interrelations between these structures.
We introduce the notion of a partial corepresentation of a given Hopf algebra $H$ over a coalgebra $C$ and the closely related concept of a partial $H$-comodule. We prove that there exists a universal coalgebra $H^{par}$, associated to the original Hopf algebra $H$, such that the category of regular partial $H$-comodules is isomorphic to the category of $H^{par}$-comodules. We introduce the notion of a Hopf coalgebroid and show that the universal coalgebra $H^{par}$ has the structure of a Hopf coalgebroid over a suitable coalgebra.
We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2-dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf alge…
The relationship between comodules of a coring and flat connections is reviewed. In particular we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in one-to-one correspondence with entwining structures, and their comodules are entwined modules. These include Yetter-Drinfeld and anti-Yetter-Drinfeld modules and their generalisations, hence all the modules of interest to Hopf-cyclic cohomology. In this way the interpretation of the latter as modules with flat connections [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential …
Let H be a finite-dimensional quasibialgebra. We show that H is a quasi-Hopf algebra if and only if the category of its finite-dimensional left modules is rigid if and only if a structure theorem for Hopf modules over H holds. We also show that a dual structure theorem for Hopf modules over a coquasibialgebra H holds if and only if the category of finite-dimensional right H-comodules is rigid; this is not equivalent to H being a coquasi-Hopf algebra.
We give a detailed comparison between the notion of a weak Hopf algebra (also called a quantum groupoid by Nikshych and Vainerman), and that of a $\times_R$-bialgebra due to Takeuchi (and also called a bialgebroid or quantum (semi)groupoid by Lu and Xu). A weak bialgebra is the same thing as a $\times_R$-bialgebra in which $R$ is Frobenius-separable. We extend the comparison to cover module and comodule theory, duality, and the question when a bialgebroid should be called a Hopf algebroid.
We introduce and study non-abelian cohomology sets of Hopf algebras with coefficients in Hopf comodule algebras. We prove that these sets generalize as well Serre's non-abelian group cohomology theory as the cohomological theory constructed by the authors in a previous article. We establish their functoriality and compute explicit examples. Further we classify Hopf torsors.
As is known to all, Hopf-Galois objects have a significant research value for analyzing tensor categories of comodules and classification questions of pointed Hopf algebras, and are natural generalizations of Hopf algebras with a Galois-theoretic flavour. In this paper, we mainly prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, and introduce the conception of Poisson Hopf-Galois algebras, and establish the relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. Moreover, we study Poisson Hopf-Galois structures on Poisson polynomi…
We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to equivariant Morita equivalence. Specifically, any indecomposable exact module categories is equivalent to the category of finite-dimensional modules over a left comodule algebra. This is an alternative approach to the results of Etingof and Ostrik. For this, we study the stabilizer introduced by Yan and Zhu and show that it coincides with the internal Hom. We a…
We introduce the concept of Hopf-Galois system, a reformulation of the notion of Galois extension of the base field for a Hopf algebra. The main feature of our definition is a generalization of the antipode of an ordinary Hopf algebra. The main application of Hopf-Galois systems is the construction of monoidal equivalences between comodule categories over Hopf algebras.
In this work we define partial (co)actions on multiplier Hopf algebras, we also present examples and properties. From a partial comodule coalgebra we construct a partial smash coproduct generalizing the constructions made by the L. Delvaux, E. Batista and J. Vercruysse.
We investigate criteria for algebra extensions that are of Galois type with respect to the coaction of a Hopf algebra or, more generally, a one-sided quotient of a Hopf algebra, or with respect to an entwining. We study the module- and comodule-theoretical properties of such extensions.
We introduce partial (co)actions of a Hopf algebra $H$ on an algebra. To this end, we introduce first the notion of lax coring, generalizing Wisbauer's notion of weak coring. We also have the dual notion of lax ring. Several duality results are given, and we develop Galois theory for partial $H$-comodule algebras.
We clarify the relation between the `bosonisation' construction (due to the author) which can be used to turn a Hopf algebra $B$ in the category of $H$-modules or $H$-comodules into an equivalent ordinary Hopf algebra, and a version of Radfords theorem (also due in this form to the author) which does the same for $B$ in the category of crossed $H$-modules. We also comment on reconstruction from the category of $B$-comodules.
Let $k$ be a commutative ring, $H$ a faithfully flat Hopf algebra with bijective antipode, $A$ a $k$-flat right $H$-comodule algebra. We investigate when a relative Hopf module is projective over the subring of coinvariants $B=A^{{\rm co}H}$, and we study the semisimplicity of the category of relative Hopf modules.
A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P->Pi whose kernels intersect to zero, P is principal if and only if all Pi's are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above d…
We show that the category of partial comodules over a Hopf algebra $H$ is comonadic over ${\sf Vect}_k$ and provide an explicit construction of this comonad using topological vector spaces. The case when $H$ is finite dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connected linear algebraic group does not admit partiality.
We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order of $G(A)$ are $>7$. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.
The paper presents applications of Toeplitz algebras in Noncommutative Geometry. As an example, a quantum Hopf fibration is given by gluing trivial U(1) bundles over quantum discs (or, synonymously, Toeplitz algebras) along their boundaries. The construction yields associated quantum line bundles over the generic Podles spheres which are isomorphic to those from the well-known Hopf fibration of quantum SU(2). The relation between these two versions of quantum Hopf fibrations is made precise by giving an isomorphism in the category of right U(1)-comodules and left modules over the C*-algebra of…
Let A be an H-Galois extension of B. If M is a Hopf bimodule then HH.(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C:=H/[H,H]. Given an injective left C-comodule V, we denote the cotensor product of M and V by N. Our aim is to investigate the relationship between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N) on the other hand. The roots of this problem can be found in Lorenz's work on the descent of Hochschild homology of centrally Galois extensions, where HH.(A)^G, the subspace of invariant cycles with respect to the…
Let A be a comodule algebra for a finite dimensional Hopf algebra K over an algebraically closed field k, and let A^K be the subalgebra of invariants. Let Z be a central subalgebra in A, which is a domain with quotient field Q. Assume that Q\otimes_Z A is a central simple algebra over Q, and either A is a finitely generated torsion-free Z-module and Z is integrally closed in Q, or A is a finite projective Z-module. Then we show that A and Z are integral over the subring of central invariants Z\cap A^K. More generally, we show that this statement is valid under the same assumptions if Z is a re…
We construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg bialgebroid of a cleft Hopf Galois extension under the condition that the corresponding Hopf algebra is cocommutative and the image of the unital cocycle corresponding to this cleft Hopf Galois extension belongs to the centre of the coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann Schauenburg bialgebroid corresponding to a Hopf Galois extension without cocycle, where comodule algebra is an ordinary smash product of the…
We define comodule algebras and Galois extensions for actions of bialgebroids. Using just module conditions we characterize the Frobenius extensions that are Galois as depth two and right balanced extensions. As a corollary, we obtain characterizations of certain weak and ordinary Hopf-Galois extensions without reference to action in the hypothesis.
We generalize the Hopf algebras of free quasisymmetric functions, quasisymmetric functions, noncommutative symmetric functions, and symmetric functions to certain representations of the category of all finite Coxeter systems and its dual category. We investigate their connections with the representation theory of 0-Hecke algebras of finite Coxeter systems. Restricted to type B and D we obtain dual graded modules and comodules over the corresponding Hopf algebras in type A.
We introduce Galois Theory for Hopf-Galois Extensions proving existence of a Galois connection between subalgebras of an H-comodule algebra and generalised quotients of the Hopf algebra H. Moreover, we show that these quotients Q which define Q-Galois extension are the closed elements of our Galois connection. We generalise important results of Hopf--Galois Theory of M. Masuoka and H.-J. Schneider by showing that there is a bijective correspondence between right ideals coideals and right coideal subalgebras of any finite dimensional Hopf algebra and we reformulate the still open problem in the…
We find a new braided Hopf structure for the algebra satisfied by the entries of the braided matrix $BSL_q(2)$. A new nonbraided algebra whose coalgebra structure is the same as the braided one is found to be a two parameter deformed algebra. It is found that this algebra is not a comodule algebra under adjoint coaction. However, it is shown that for a certain value of one of the deformation parameters the braided algebra becomes a comodule algebra under the coaction of this nonbraided algebra. The transmutation of the nonbraided algebra to the braided one is constructed explicitly.
We show that the differential complex $Ω_{B}$ over the braided matrix algebra $BM_{q}(N)$ represents a covariant comodule with respect to the coaction of the Hopf algebra $Ω_{A}$ which is a differential extension of $GL_{q}(N)$. On the other hand, the algebra $Ω_{A}$ is a covariant braided comodule with respect to the coaction of the braided Hopf algebra $Ω_{B}$. Geometrical aspects of these results are discussed.
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.
We introduce the analogue of Manin's universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras $P$ and $U$, where $U$ is finite dimensional, we construct a Poisson algebra $\mathcal{B}(P,\, U)$ together with a Poisson algebra homomorphism $ψ_{\mathcal{B}(P,\,U)} \colon P \to U \otimes \mathcal{B}(P,\, U)$ satisfying a suitable universal property. $\mathcal{B}(P,\, U)$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $P=U$ is a finite dimensional Poisson alg…
The cyclic (co)homology of Hopf algebras is defined by Connes and Moscovici [math.DG/9806109] and later extended by Khalkhali et.al [math.KT/0306288] to admit stable anti-Yetter-Drinfeld coefficient module/comodules. In this paper we will show that one can further extend the cyclic homology of Hopf algebras with coefficients non-trivially. The new homology, called the bialgebra cyclic homology, admits stable coefficient module/comodules, dropping the anti-Yetter-Drinfeld condition. This fact allows the new homology to use bialgebras, not just Hopf algebras. We will also give computations for b…
We introduce the notion of characters of comodules over coribbon Hopf algebras. The case of quantum groups of type $A_n$ is studied. We establish a characteristic equation for the quantum matrix and a q-analogue of Harish-Chandra- Itzykson-Zuber integral
In previous work, Eli Aljadeff and the first-named author attached an algebra B_H of rational fractions to each Hopf algebra H. The generalized Noether problem is the following: for which finite-dimensional Hopf algebra H is B_H the localization of a polynomial algebra? A positive answer to this question when H is the algebra of functions on a finite group implies a positive answer for the classical Noether problem for the group. We show that the generalized Noether problem has a positive answer for all pointed finite-dimensional Hopf algebras over a field of characteristic zero. We actually g…
We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra $H$. If $H$ is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras. Graded symmetric algebras are also investigated.
We discuss some axioms that ensure that a Hopf algebra has its simple comodules classified using an analogue of the Borel-Weil construction. More precisely we show that a Hopf algebra having a dense big cell satisfies the above requirement. This method has its roots in the work of Parshall and Wang in the case of q-deformed quantum groups GL and SL. Here we examine the example of universal cosovereign Hopf algebras, for which the weight group is the free group on two generators.
For a quasi-Hopf algebra $H$, a left $H$-comodule algebra $\mf{B}$ and a right $H$-module coalgebra $C$ we will characterize the category of Doi-Hopf modules ${}^C{\cal M}(H)_{\mf{B}}$ in terms of modules. We will also show that for an $H$-bicomodule algebra $\mb{A}$ and an $H$-bimodule coalgebra $C$ the category of generalized Yetter-Drinfeld modules ${}_{\mb{A}}{\cal YD}(H)^C$ is isomorphic to a certain category of Doi-Hopf modules. Using this isomorphism we will transport the properties from the category of Doi-Hopf modules to the category of generalized Yetter-Drinfeld modules.
