The Semigroup Structure of Left Clifford Semirings

In this paper,we generalize Clifford semirings to left Clifford semirings by means of the so-called band semirings.We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.     

一类商代数上的双-Frobenius代数结构

设C[X]为复数域上的一元多项式代数,I为n+1次Dickson多项式E_(n+1)(X)生成的C[X]的理想,C[X]/I为商代数.证明了商代数C[X]/I既是Frobenius代数,又是Frobenius余代数.进一步,该商代数在恒等对极下还是双-Frobenius代数.     

Varieties of Two-Step Solvable Algebras of Type (1, 1)

The structure of the set of all non-nilpotent subvarieties of the variety of two-step solvable algebras of type (1,1) is studied. An additive basis of a free metabelian (1,1)-algebra is constructed. It is proved that any identity in a non-nilpotent metabelian (1,1)-algebra of degree ≥6 is a consequence of four defining relations.     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

半群OI_n(k)的秩

设OI_n是[n]上的保序严格部分一一变换半群.对任意1≤k≤n-1,研究半群OI_n(k)={α∈OI_n:(■x∈dom(α))x≤k■xα≤k}的秩,证明了半群OI_n(k)的秩为n+1.     

半群O_n(k)的秩

设O_n是有限链[n]上的保序变换半群.对任意1≤k≤n-1,研究半群O_n(k)={α∈O_n:(x∈[n]x≤k→xα≤k}的秩和幂等元秩,证明了半群O_n(k)的秩为2n-3.进一步,得到了半群O_n(k)(2≤k≤n-1)的幂等元秩为n和半群O_n(1)的幂等元秩为n-1.     

Invariants for Automorphisms of the Underlying Algebras Relative to Lie Algebras of Cartan Type

Let X denote a finite or infinite dimensional Lie algebra of Cartan type W, S, H or K over a field of characteristic p 〉 3. In this paper it is proved that certain filtrations of the underlying algebras are invariant under the admissible groups relative to Lie algebras of Cartan type X.     

Ixr(a)的有限型Ixor(a)的未定Weyl群

本文首先给出Kac-Moody代数IXr(a)的有限型IC or (a)的未定Weyl群的定义,然后对a≥5证明了不定型李代数IXr(a)的Weyl群W同构于有限型IXr(a)的未定Weyl群.     

Application of Vertex Algebras to the Structure Theory of Certain Representations Over the Virasoro Algebra

In this paper, we discuss the structure of the tensor product \(V_{\alpha,\beta }^{\prime}\otimes L(c,h)\) of an irreducible module from an intermediate series and irreducible highest-weight module over the Virasoro algebra. We generalize Zhang’s irreducibility criterion from Zhang (J Algebra 190:1–10, 1997), and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operators for modules over vertex operator algebra yields reducibility of \(V_{\alpha ,\beta}^{\prime}\otimes L(c,h)\) , which is a completely new point of view to this problem. As an example, the complete structure of the tensor product with minimal models c?=???22/5 and c?=?1/2 is presented.     

Discrete Valuations Extend to Certain Algebras of Quantum Type

So-called “quantized” algebras are popular objects of study in non-commutative algebra. Usually such algebras are either positively graded Ore domains R with R ₀ = K a field and R = K[R ₁], R ₁ being a finite dimensional K vectorspace, or else filtered rings having a ring of forementioned type for its associated graded ring. We show that every discrete valuation of K extends to a valuation, in the sence of O. Schilling (cf. [S]), of the skewfield of fractions, Δ = Q_cl (R), of the Ore domain R (Proposition 2.3. and Corollary 2.8.). Such extension property has long been known to fail for finite dimensional skewfields over K; its validity in the case of several quantized algebras may be viewed as a consequence of the rigidity of their defining relations. Our result opens the door for a more arithmetical study of Δ e.g. in case K is a numberfield or an algebraic function field of a curve; for an application in this direction we refer to a first version of some divisor calculus started in [VW].     

n+1维n-Lei代数的导子代数

本文给出了n 1维n-Lie代数的导子代数的结构，以及导子的具体表示形式．     

On the Rank of the Semigroup TE(X)

${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5.     

Hochschild (Co)homology of Exterior Algebras

The minimal projective bimodule resolutions of the exterior algebras are explicitly constructed. They are applied to calculate the Hochschild (co)homology of the exterior algebras. Thus the cyclic homology of the exterior algebras can be calculated in case the underlying field is of characteristic zero. Moreover, the Hochschild cohomology rings of the exterior algebras are determined by generators and relations.     

Bourgain空间X~(s,b)及其在KdV型方程中的应用(英文)

本文总结了使用Bourgain空间技术研究KdV型方程初值问题的局部适定性和整体适定性方面所取得的结果.     

Structure of <Emphasis Type=Italic>H</Emphasis>-Semiprime Artinian Algebras

Let H be a Hopf algebra over a field. It is proved that every H-semiprime right artinian left H-module algebra A is quasi-Frobenius and H-semisimple. If H grows slower than exponentially, then all H-equivariant A-modules are shown to be A-projective. With the additional assumption that H is cosemisimple it is proved that the Jacobson radical of any right artinian left H-module algebra is stable under the action of H.     

Some Algebras Similar to the Enveloping Algebra of s1(2)

Abstract

Consider the algebra R = ?[A,B,H] subject to relations

where and . We study these algebras for different f and ç. We study highest weight modules and finite dimensional simple modules. We introduce the categories ^p with p ≥ 1, with an interesting decomposition in categories ^p _I and ^p _II . We also consider the BGG-category and develop new techniques towards the study of .     

The Lanchester square-law model extended to a (2,2) conflict

A natural extension of the Lanchester (1,1) square-law modelis the (M,N) linear model in which M forces oppose N forceswith constant attrition rates. The (2,2) model is treated fromboth direct and inverse viewpoints. The inverse problem meansthat the model is to be fitted to a minimum number of observedforce levels, i.e. the attrition rates are to be found fromthe initial force levels together with the levels observed attwo subsequent times. An approach based on Hamiltonian dynamicshas enabled the authors to derive a procedure for solving theinverse problem, which is readily computerized. Conflicts inwhich participants unexpectedly rally or weaken must be excluded.