Oriented quantum coalgebra structure on the tensor product of an oriented quantum coalgebra with itself

Oriented quantum algebras （coalgebras） are generalizations of quasitriangular Hopf algebras （coquasitriangular Hopf algebras） and account for regular isotopy invariants of oriented 1-1 tangles, oriented knots and links. Let （H, or, D, U） be an oriented quantum coalgebra over the field k. Then （H×H, φ, D×D, U× U） is a trivial oriented quantum coalgebra structure on the tensor product of coalgebra H with itself, where φ （a × b, c × d） = σ-（ a, c）σ （b, d）. This paper presents the oriented quantum coalgebra structure （ H×H, σ, D×D, U× U） on coalgebra H× H, where σ（ a × b, c× d） = σ ^-1 （ d1, a1 ） σ（ a2, c1 ） σ^-1 （ d2, b1 ） σ（ b2, c2 ）. So a nontrivial oriented quantum coalgebra structure is obtained and it is dual to Radford＇s results in the paper ＂On the tensor product of an oriented quantum algebra with itself＂ published in 2007. Theoretically, the results of this paper are important in constructing the invariants of oriented knots and links.

The Structure Theorem for Module Coalgebras

This paper,mainly gives the structure theorem for module coalgebras by a kind of new method,and deletes the condition that the antipode S of the Hopf algebra H is bijective.

Group Twisted Tensor Bipro ducts over Hopf Group Coalgebras

In this paper, we introduce the concept of a group twisted tensor biproduct and give the necessary and su？cient conditions for the new object to be a Hopf group coalgebra.

On the Decomposition of Hopf Module Coalgebra

In this paper, the module coradical is introduced. From this, we show that any Hopf module coalgebra which is local finite can be uniquely decomposed into a direct sum of its indecomposable components. This result is a generalization of the decompostion theorem of coalgebra.

On the Structure of the B-cocleft Module Coalgebra

In this paper we develope the notions of crossed coproduct of Hopf algebras and study an equivalent theorem of generalized crossed coproduct of H weakly comodule coalgebras and H module coalgebras. The main result is to prove a structure theorem about B cocleft H module coalgebras.

THE CORRESPONDENCE THEOREM FOR COALGEBRAS

In this paper,the correspondence theorem for the usual algebraic systems is generalized to coalgebra.Moreover,one example,which illustrate the preimage of a subcoalgebra need not to be a subcoalgebra for general coalgebra homomorphism,is given.

Constructing the Cocyclic Structures for Crossed Coproduct Coalgebras

In this paper, we construct a cocylindrical object associated to two coalgebras and a cotwisted map. It is shown that there exists an isomorphism between the cocyclic object of the crossed coproduet coalgebra induced from two coalgebras with a cotwisted map and the cocyclic object related to the diagonal of the cocylindrical object.

（＊）-Serial coalgebras

In this paper, we introduce the notion of （＊）-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of （＊）-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be （＊）-serial.

Cylinder Coalgebras and Cylinder Coproducts for Quasitriangular Hopf Algebras

This paper introduces the concepts of cylinder coalgebras and cylinder coproducts for quasitriangular bialgebras, and points out that there exists an anti-coalgebra isomorphism （H,△^-）≌ （H,△^-）, where （H, △^-） is the cylinder coproduct, and （H,△^-） is the braided coproduct given by Kass. For any finite dimensional Hopf algebra H, the Drinfel＇d double （D（H）,△^-D（H）） is proved to be the cylinder coproduct. Let （H, H, R） be copaired Hopf algebras. If R ∈ Z（H×H） with inverse R-1 and skew inverse R, then the twisted coalgebra （H^R）^R-1 is constructed via twice twists, whose comultiplication is exactly the cylinder coproduct. Moreover, for any generalized Long dimodule, some solutions for Yang-Baxter equations, four braid pairs and Long equations are constructed via cylinder twists.

Morita theorems for Hopf comodule coalgebras

Morita context has been used to study algebra structure and category equivalenee sinceMorita theory was established in 1958. For an H-module algebra A, Cohen and Fischmanconstructed in 1986 a Morita contex [A^H,A,A,A#H] under the additional assumption thatH was unimodular and used the Morita context to study the Smash product A#H. In1990, Cohen, Fischman and Montgomery showed that fix ring A^H and smash product

Dorroh extensions of algebras and coalgebras

We study Dorroh extensions of algebras and Dorroh extensions of coalgebras.Their structures are described.Some properties of these extensions are presented.We also introduce the finite duals of algebras and modules which are not necessarily unital.Using these finite duals,we determine the dual relations between the two kinds of extensions.

正规三角矩阵余代数上的余导子

设C和D为余代数,M为C-D-双余模.该文给出了正规三角矩阵余代数T=(CM0D)上余导子的具体结构,并建立T的余导子与C、D和M上相关概念的具体关系.

A Duality Theorem for Hopf Comodule Coalgebras

R. J. Blattner and S. Montgomery have proved the duality theorem of Hopf module algebras in Ref. [1]. This theorem contains duality for crossed product of von Neumann algebras. In 1977, R. K. Molnar introduced the concept of Hopf comodule coalgebras which is a dual notation of Hopf module algebra, and discussed their properties. However, the duality theorem of Hopf comodule coalgebras has not been proved yet. In this note we shall deal with this situation by defining the

The Structure of Cocleft H-Module Coalgebras

Y. Doi has investigated the H-comodule algebra since 1983. The structure theorem of cleft H-comodule algebra was given by Y. Doi and M. Takeuchi in 1986, i. e. if A is a cleft H-comodule algebra then A≌A0#σH as algebra. The dual notion of H-comodule algebra is H-module coalgebra. Y. Doi also studied the H-module coalgebra. However, he did not obtain the structure theorem of cocleft H-module coalgebra. In this note we introduce crossed coproducts which are the dual notion

Relation between decomposition of comodules and coalgebras

Let M be a C-comodule. It is clear that M and C can both be decomposed into a direct sum of the indecomposable subcoalgebras of C and subcomodules of M. The relation between the two decompositions is given

Using coalgebras and the Giry monad for interpreting game logics a tutorial

The stochastic interpretation of Parikh＇s game logic should not follow the usual pattern of Kripke models, which in turn are based on the Kleisli morphisms for the Giry monad, rather, a specific and more general approach to proba- bilistic nondeterminism is required. We outline this approach together with its probabilistic and measure theoretic basis, in- troducing in a leisurely pace the Giry monad and its Kleisli morphisms together with important techniques for manipu- lating them. Proof establishing specific techniques are given, and pointers to the extant literature are provided. After working through this tutorial, the reader should find it easier to follow the original literature in this and related areas, and it should be possible for her or him to appreciate measure theoretic arguments for original work in the areas of Markov transition systems, and stochastic effectivity func- tions.

Quasitriangular Turaev Group Coalgebras and Radford＇s Biproduct

In this paper, we construct a new example of Hopf group coalgebras by con- sidering Radford＇s biproduct Hopf algebra in the Turaev category. Furthermore, we find some sufficient and necessary conditions for such Radford＇s biproduct Hopf algebra to admit quasitriangulax structures in the sense of Turaev group coalgebras.

On Gorenstein coalgebras

We investigate the comodule representation category over the Morita-Takeuchi context coalgebra F and study the Gorensteinness of F. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra F and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic.

Valued Gabriel quiver of a wedge product and semiprime coalgebras

We describe the valued Gabriel quiver of a wedge product of coalgebras and study the category of comodules of a semiprime coalgebra. In particular, we prove that any monomial semiprime k-tame fc-tame coalgebra is string. We also prove a version of Eisenbud-Griffith theorem for coalgebras, namely, any hereditary semiprime strictly quasi-finite coalgebra is serial.

Duality theorem for smash coproduct over quantum groupoids

The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.