Lie comodules and the constructions of Lie bialgebras

In this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.

Equivalences between Categories of Modules and Categories of Comodules

We show the close connection between apparently different Galois theories for comodules introduced recently in [J. Gomez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, Algebr. Represent. Theory, 10 （2007）, 271 306] and [Wisbauer, On Galois comodules, Comm. Algebra 34 （2006）, 2683-2711]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.

Some remarks on cotilting comodules

We study cotilting comodules and f-cotilting comodules and give a description of localization of f-cotilting comodules and classical tilting comodules. First, we introduce T-cotilting injective comodules and their dimensions which are important for researching cotilting comodules. Then we characterize the localization in f-cotilting comodules, finitely copresented comodules, and classical tilting comodules. In particular, we obtain a localizing property about finitely copresented comodules.

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

Decomposition of Comodules

It has been proved, using a special approach, that any comodules over a field can be uniquely decomposed into the direct sum of closed indecomposable subcomodules.

Recent Developments in Morita Equivalence: Part Two

Morita equivalence was established by Morita in the late 1950’s. In part one most of the recent developments in this theory on the categories of modules over rings were reviewed and some results were presented. Here this paper discusses those on categories of comodules over coalgebras, these results are due to Ling, Takeuchi and Wang.

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.

THE UNIVERSAL DUAL COMODULE OF MODULE IN HOPF ALGEBRAS

The purpose of this paper is to present some dual properties of dual comodule. It turns out that dual comodule has universal property (cf.Theorem 2). Since (( )*,()°) is an adjoint pair (cf.Theorem 3), some nice properties of functor ( )° are obtained. Finally Theoram 4 provides that the cotensor product is the dual of the tensor product by (M (?)A N)°≌M°□A°N°. Moreover, the result Hom(M,JV)≌ComA°(N°,M°) is proved for finite related modules M, N over a reflexive algebra A.

THE FIELD ALGEBRA IN HOPF SPIN MODELS DETERMINED BY A HOPF*-SUBALGEBRA AND ITS SYMMETRIC STRUCTURE

Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More precisely, we consider the quantum double D(H, H_(1)) as the bicrossed product of the opposite dual Hopˆ of H and H1 with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H1 and Ĥ we define the observable algebra AH1. Then using a comodule action of D(H, H1) on AH1, we obtain the field algebra FH1, which is the crossed product AH1⋊D(H,H_(1)), and show that the observable algebra AH1 is exactly a D(H, H1)-invariant subalgebra of FH1. Furthermore, we prove that there exists a duality between D(H, H1) and AH1, implemented by a*-homomorphism of D(H, H_(1)).

QUANTUM COMODULE APPROACH THE SOLUTION OF THE QUANTUM YANG-BAXTER EQUATION

In this paper, the relationship between solutions of the Quantum Yang-Baxter Equation and quantum comodules, and some properties of the quantum comodule category are characterized here. These results make it possible to give some set-theoretical solutions of the Quantum Yang-Baxter Equation.

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 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 tensor algebras of Yetter-Drinfeld module

The antipode of a Yetter-Drinfeld Hopf algebra is an anti-algebra and anti-coalgebra map is proved. It is also proved that the tensor algebra of Yetter-Drinfeld Hopf module is a Yetter-Drinfeld Hopf algebra.

A duality theorem of crossed coproduct for Hopf algebras

A duality theorem for Hopf crossed coproduct is proved. This theorem plays a role similar to that appearing in the work of Koppinen (which generalized the corresponding results of group grraded ring).

Floating-gate photosensitive synaptic transistors with tunable functions for neuromorphic computing

Synaptic devices that merge memory and processing functions into one unit have broad application potentials in neuromorphic computing, soft robots, and humanmachine interfaces. However, most previously reported synaptic devices exhibit fixed performance once been fabricated,which limits their application in diverse scenarios. Here, we report floating-gate photosensitive synaptic transistors with charge-trapping perovskite quantum dots(PQDs) and atomic layer deposited(ALD) Al_(2)O_(3) tunneling layers, which exhibit typical synaptic behaviors including excitatory postsynaptic current(EPSC), pair-pulse facilitation and dynamic filtering characteristics under both electrical or optical signal stimulation. Further, the combination of the high-quality Al2O3 tuning layer and highly photosensitive PQDs charge-trapping layer provides the devices with extensively tunable synaptic performance under optical and electrical co-modulation. Applying light during electrical modulation can significantly improve both the synaptic weight changes and the nonlinearity of weight updates, while the memory effect under light modulation can be obviously adjusted by the gate voltage.The pattern learning and forgetting processes for "0" and "1"with different synaptic weights and memory times are further demonstrated in the device array. Overall, this work provides synaptic devices with tunable functions for building complex and robust artificial neural networks.

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

（＊）-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.

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

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.