We introduce the notion of idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras, and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming bijective antipode.
The characterization of $H$-prime radical is given in many ways. Meantime, the relations between the radical of smash product $R # H$ and the $H$-radical of Hopf module algebra $R$ are obtained.
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.
We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question of A. Agore on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a $B$-(co)module, for a bialgebra $B$.
Let H be a finite-dimensional pivotal and unimodular Hopf algebra over a field k. It was shown in [BBGa] that the projective tensor ideal in H-mod admits a unique non-degenerate modified trace, a natural generalisation of the categorical trace. This paper provides an extension of this result to a much more general setting. We first extend the notion of the modified trace to the so-called module trace for a given k-linear module category M over a pivotal category C equipped with a module endofunctor. We provide a non-trivial class of examples of such module traces. In particular, we show that a…
In this article an interpretation and a proof of some classical \\theorems in analysis on the integration of analytic vectors fields are derived from the algebraic method of realization of bialgebras which are constructed with the data of a linear application from a coalgebra into the algebra of right (or left) invariant operators on an approximated coalgebra. The results are obtained from these general algebraic construction and theorems by introducing the more restrictive notion of induced module. Then the associated envelopping bialgebra is defined and naturally belongs to the dual of the t…
For a given entwining structure $(A,C)_ψ$ involving an algebra $A$, a coalgebra $C$, and an entwining map $ψ: C\otimes A\to A\otimes C$, a category $\M_A^C(ψ)$ of right $(A,C)_ψ$-modules is defined and its structure analysed. In particular, the notion of a measuring of $(A,C)_ψ$ to $(\tA,\tC)_\tpsi$ is introduced, and certain functors between $\M_A^C(ψ)$ and $\M_\tA^\tC(\tpsi)$ induced by such a measuring are defined. It is shown that these functors are inverse equivalences iff they are exact (or one of them faithfully exact) and the measuring satisfies a certain Galois-type condition. Next, l…
The paper presents a detailed description of duality for braided algebras, coalgebras, bialgebras, Hopf algebras and their modules and comodules in the infinite setting. Assuming that the dual objects exist, it is shown how a given braiding induces compatible braidings for the dual objects, and how actions (resp. coactions) can be turned into coactions (resp. actions) of the dual coalgebra (resp. algebra), with an emphasis on braided bialgebras and their braided (co)module algebras.
Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems, known for Doi-Hopf modules are extended to the case of entwined modules.
We construct the $H$-von Neumann regular radical for $H$-module algebras and show that it is an $H$-radical property. We obtain that the Jacobson radical of twisted graded algebra is a graded ideal. For twisted $H$-module algebra $R$, we also show that $r_{j}(R#_σH)= r_{Hj}(R)#_σH$ and the Jacobson radical of $R$ is stable, when $k$ is an algebraically closed field or there exists an algebraic closure $F$ of $k$ such that $r_j(R \otimes F) = r_j(R) \otimes F$, where $H$ is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf algebra over $k$. In particular, we answ…
The unified product was defined in \cite{am3} related to the restricted extending structure problem for Hopf algebras: a Hopf algebra $E$ factorizes through a Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1\in H$ if and only if $E$ is isomorphic to a unified product $A \ltimes H$. Using the concept of normality of a morphism of coalgebras in the sense of Andruskiewitsch and Devoto we prove an equivalent description for the unified product from the point of view of split morphisms of Hopf algebras. A Hopf algebra $E$ is isomorphic to a unified product $A \ltimes H$ if and only if there …
For a coalgebra $C_k$ over field $k$, we define the "coalgebra extension problem" as the question: what multiplication laws can we define on $C_k$ to make it a bialgebra over $k$? This paper answers this existence-uniqueness question for certain coalgebras called "circulant coalgebras". We begin with the trigonometric coalgebra, comparing and contrasting with the group-(bi)algebra $k[\mathbb Z/2]$. This leads to a generalization, the dual coalgebra to the group-algebra $k[\mathbb Z/p]$, which we then investigate. We show connections with other questions, motivating us to answer to the coalgebr…
The aim of this paper is to construct comodules of $U_q(sl_2)$ and modules of $SL_q(2)$ via quiver, where $q$ is not a root of unity. By embedding the quantized algebra $U_q(sl_2)$ into the path coalgebra $k\mathcal{D}^c$, where $\mathcal{D}$ is the Gabriel quiver of $U_q(sl_2)$ as a coalgebra, we obtain a basis of $U_q(sl_2)$ in terms of combinations of paths in the quiver $\mathcal{D}$; this special basis enable us to describe the category of $U_q(sl_2)$-comodules by certain representations of $\mathcal{D}$; and this description further permits us to construct a class of modules of the quant…
For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ is positive (and lifts to a positive map on the full $C^*$ completion on $H$) if and only if $A$ is invariant under the squared antipode of $H$. The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding to $A\to H$ by means of a fiber functor fro…
This article concerns a generalization of the Temperley-Lieb algebra, important in applications to conformal field theory. We call this algebra the valenced Temperley-Lieb algebra. We prove salient facts concerning this algebra and its representation theory, which are both of independent interest and used in our subsequent work, where we uniquely and explicitly characterize the monodromy invariant correlation functions of certain conformal field theories.
After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter--Drinfel'd modules over a bicoalgebroid. It is proved that the Yetter--Drinfel'd category is monoidal and pre--braided just as in the case of bialgebroids, and is embedded into the one--sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzezinski and Militaru [2] and Balint and Slachanyi [1], originally app…
The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a Hom-bialgebra with bijective structure map provide solutions of the Hom-Yang-Baxter equation. The category $_H^H{\mathcal YD}$ of Yetter-Drinfeld modules with bijective structure maps over a Hom-bialgebra H with bijective structure map can be organized, in two different ways, as a quasi-braided pre-tensor category. If H is quasitriangular (respectively coquasitri…
We study monoidal structures on the category of (co)modules over a weak bialgebra. Results due to Nill and Szlachányi are unified and extended to infinite algebras. We discuss the coalgebra structure on the source and target space of a weak bialgebra.
A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the `base algebras') are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier bialgebra are shown to constitute a monoidal category via the (co)module tensor product over the base algebra. The relation to Van Daele and Wang's (regular and arbitrary) weak multiplier Hopf algebra is discussed.
