Normal Forms of Modules over Admissible Algebras with Formal Two-ray Modules

The aim of the paper is to classify the indecomposable modules and describe the Auslander-Reiten sequences for the admissible algebras with formal two-ray modules.     

弱Hopf量子Yang-Baxter H-模基本定理(英文)

本文引入弱Hopf量子Yang-Baxter模概念.利用弱Hopf模基本定理的方法,获得了弱Hopf量子Yang-Baxter模基本定理,进一步还得到了相关Hopf模基本定理.     

Lie Bialgebras of a Family of Lie Algebras of Block Type

Lie bialgebra structures on a family of Lie algebras of Block type are shown to be triangular coboundary.     

Modules over Operator Algebras and Its Module Maps

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Representations of a Class of Associative Algebras Related to the Quantum Torus

1 Introduction The mitivation of this paper comes from the work of[5].We kmow that vertex algebratheory is one of the importans parts in the study of infinte dimensional Lie theory,while thelattice vertex algebras form one of the most important and fundamental classes of vertexalgebras.In they studied the representation theory for certain” half lattice”vertex algebra     

Gorenstein-Projective Modules over T_（m,n）（A）

A new class of Gorenstein algebras T _m,n (A) is introduced, their module categories are described, and all the Gorenstein-projective T _m,n (A)-modules are explicitly determined.     

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.     

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over Twisted Heisenberg-Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.     

重模代数和量子Yang-Baxter模代数的量子化

In this paper, we mainly construct quantization of dimodule algebras and quantum Yang-Baxter H-module algebras, and give some results of smash products and braided products.     

弱斜配对双代数和弱相关Long双代数

该文在弱双代数$H$上给出了扭曲积$(H^\sigma,\cdot_\sigma)$成为弱双代数的充分必要条件.设$[B, H, \tau]$是一个弱斜配对, 并且$\tau$可逆,则在某个条件下弱双交叉积$B\bowtie_\tau H$是一个弱双代数. 如果$(B,H, \sigma)$是弱相关Long双代数, 并且$\sigma$可逆,则弱双交叉积$B^{OP}\bowtie_\sigma H$可以被构造. 它的乘法是:$(x\otimes h)(y\otimes g)=\Sigma\sigma(y_1, h_1)y_2x\otimes h_2g\sigma^{-1}(y_3, h_3),$ 特别地, 如果$(B, H,\sigma)$是相关Long双代数, 则$(B^{OP \bowtie_\sigma H,\beta)$是Long双代数当且仅当对任意$b, d\in B^{OP}; g, \ell\in H$,$\Sigma\sigma^{-1}(b, g_2\ell)\sigma(d, g_1)=\Sigma\sigma^{-1}(b,\ell g_1)\sigma(d, g_2),$ 其中$B$为$H$的子Hopf代数,$\beta$定义为$\beta(b\bowtie_\sigma h\otimes c\bowtie_\sigma g)=\varepsilon_H(h)\varepsilon_{B^{OP}}(c)\sigma^{-1}(b, g).$ 对于Sweedler 4维Hopf代数$H$, 作者给出一个例子说明:此弱双交叉积$(B^{OP}\bowtie_\sigma H, \beta)$不仅是一个Long双代数,而且是一个非可换和非余可换的8维Hopf代数. 最后, 设$B,H$都是弱双代数, $\sigma: B\otimes H\rightarrow k$是一个线性映射, 作者给出了$(B,\sigma,\leftharpoonup, \Delta_B)$是弱相关右$(H, B)$ -重模代数的充分必要条件.     

Lie Bialgebras of Generalized Loop Virasoro Algebras

The first cohomology group of generalized loop Virasoro algebras with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is used to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. The authors generalize the results to generalized map Virasoro algebras.     

量子代数模的张量积与不可分解模

令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子代数.k是特征为零的代数闭域,A→k(v(?)ξ)是环同态.U_k=U(?)_Ak,u_k是U_k的无穷小量子代数.令ξ是1的p次本原根.本文证明了:若有限维可积U_k模M,V中至少有一个是内射模,或者M,V中有一个模作为u_k模是平凡的,则有U_k模同构M(?)V≌V(?)M.我们还证明了:若有限维可积U_k模V作为u_k模是不可分解的,有限维可积U_k模M是不可分解的,且M|_(uk)是平凡的,则V(?)M是不可分解U_k模.令V和M是有限维可积U_k模,作为u_k模是同构的且具有单基座,本文证明V和M作为U_k模也是同构的.由此得到:不可分解内射u_k模提升为U_k模是唯一的.     

Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category . Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group the corresponding braided category of modules is identified with a full subcategory in . The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.     

Smash积,辫积和L-R Smash积的新对偶

本文主要地证明:由H-重模代数A,B构成的Smash积A#B的新对偶H(A#B)~0恰好是由重模余代数_HA~0,_HB~0构成的Smash余积_HA~0×_HB~0;如果(H,σ)是辫化Hopf代数,则新对偶_HH~0是右,左H~0-重模余代数;由量子Yang-Baxter H-模代数A,B构成的辫积AαB的新对偶(AαB)~0恰好是由量子Yang-Baxiter H-模余代数_HA~0,_HB~0构成的辫余积_HA~0×_HB~0.最后它给出由H-双模代数A构成的L-R Smash积A■H的新对偶(A■H)_H~0的正合序列。     

对合导子构造的3-李双代数与3-Pre-李代数

研究具有对合导子的3-李代数的结构,证明了具有对合导子的m-维3-李代数A存在相容的3-Pre-李代数,且在2m-维半直积3-李代数A■_(ad~*)A~*上存在局部上循环3-李双代数结构.利用对合导子构造了3-李代数A■_(ad~*)A~*上的3-李Yang-Baxter方程的解和一类3-Pre-李代数,并构造了8-维和10-维局部上循环3-李双代数.     

Extensions of Hopf Algebras and Lie Bialgebras

Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .

     

Quasidiagonal Solutions of the Yang–Baxter Equation, Quantum Groups and Quantum Super Groups

This paper answers a few questions about algebraic aspects of bialgebras, associated with the family of solutions of the quantum Yang–Baxter equation in Acta Appl. Math. 41 (1995), pp. 57–98. We describe the relations of the bialgebras associated with these solutions and the standard deformations of GL_n and of the supergroup GL(m|n). We also show how the existence of zero divisors in some of these algebras are related to the combinatorics of their related matrix, providing a necessary and sufficient condition for the bialgebras to be a domain. We consider their Poincaré series, and we provide a Hopf algebra structure to quotients of these bialgebras in an explicit way. We discuss the problems involved with the lift of the Hopf algebra structure, working only by localization.     

Cat1-Hopf Algebras and Crossed Modules

A reductive monoid M is called 𝒥-irreducible if M?{0} has exactly one minimal G × G-orbit. There is a canonical cellular decomposition for such monoids. These cells are defined in terms of idempotents, B × B-orbits, and other natural monoid notions. But they can also be obtained by the method of one-parameter subgroups. This decomposition leads to a number of important combinatorial and topologial properties of the monoid of B × B-orbits of M. In case M?{0} is rationally smooth these cells are closely related to affine spaces. They can be used to calculate the Betti numbers of a certain projective variety.