A. Ardizzoni, C. Menini, D. Stefan
We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an abelian monoidal category. Under some further conditions we show that such a cotensor coalgebra exists and satisfies a meaningful universal property. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth.
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The construction of the cotensor coalgebra for an "abelian monoidal" category $\M$ which is also cocomplete, complete and AB5, was performed in [A. Ardizzoni, C. Menini and D. Ştefan, \emph{Cotensor Coalgebras in Monoidal Categories}, Comm. Algebra, to appear]. It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra $E$ in $\M$ is filled by considering a direct limit $\widetilde{D}$ of a filtration consisting of wedge products of a subcoalgebra $D$ of $E$. The main aim of this …
Cotensor decompositions for comodules over graded vertex operator coalgebras
本文针对分级顶点算子余代数，构造可容许余模的逆步余模与余交织算子，通过泛性质定义余张量分解，并证明其复化Grothendieck群为余交换余结合余代数，且对偶于对偶顶点算子代数的融合代数。
A cotensor product A\Box_H P of an H-Hopf Galois extension A and a C-coalgebra Galois extension P, such that P is an (H,C)-bicomodule, is analyzed. Conditions are stated, when A\Box_H P is a C-coalgebra Galois extension and when there exists a strong connection on A\Box_H P. Two examples are given, in both, A and P are Matsumoto spheres, and H=C=C(U(1)).
In this article we defined and studied quasi-finite comodules, the cohom functors for coalgebras over rings. linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work characterizes equivalences between comodule categories generalizing the Morita-Takeuchi theory to coalgebras over rings. Morita-Takeuchi contexts in our setting is defined and investigated, a correspondence between strict Morita-Takeuchi contexts and equivalences of comodule categories over the involved c…
We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.
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 $B$.
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter-Drinfeld modules over Hopf group-coalgebras using the center construction.
On deformation theory of associative algebras in monoidal categories
本文通过幺半范畴和余交换余代数推广了结合代数的经典形变理论，构造了函子性范畴框架，证明了形变预层的可表示性，并识别了支配该形变的上同调理论。
Incidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner's bigraphs and crossed modules are considered.
Braided cohomology of quasi-triangular bialgebras and braided Morita invariance
引入线性幺半范畴中编织余代数的编织上链复形和编织上同调，证明在有限维条件下拟三角双代数的编织上链复形是编织Morita不变量。
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.
We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.
Braided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly $\mathbb{N}$-graded both as an algebra and as a coalgebra.
We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a cat…
For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$ admits a monoidal structure by work of Nill and Böhm-Caenepeel-Janssen, but the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras. Nevertheless, we prove that there is an isomorphism between the category o…
Let H be a bialgebra and D an H-bimodule algebra H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D\otimes H to form a bialgebra (in this case we say that (H, D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category {\cal LR}(H), whose objects are H-bimodules H-bicomodules M endowed with left-left and right-right Yetter-Drinfeld module as well as left-right and rig…
The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomology. Now let us consider a Hopf algebra A such that its Jacobson radical J is a nilpotent Hopf ideal and H:=A/J is a semisimple algebra. By using our homological results, we prove that the canonical proje…
Given an abelian k-linear rigid monoidal category V, where k is a perfect field, we define squared coalgebras as objects of cocompleted V tensor V (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If V is the category of k-vector spaces, squared (co)algebras coincide with conventional ones. If V is braided, a braided Hopf algebra can be obtained from a squared one. Reconstruction theorems give equivalence of squared co- (bi-, Hopf) algebras in V and corres…
Yang-Baxter Solutions from Categorical Augmented Racks
本文在张量范畴中解释增广架，并基于量子堆、霍普夫代数伴随作用及范畴化增广架的归纳构造，给出杨-巴克斯特方程的无穷族解和辫子幺半范畴构造。
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