COHOMOLOGY OF GRADED LIE ALGEBRAS OF CARTAN TYPE S (n, m)

Let F be an algebraically closed field of characteristic p>3. This paper studies the cohomology of graded Lie algebras S(n,m) of Cartan type over F. The author determines the structures of H~1(S(3,m), _0), where _0 is the graded S(3,m)-module which is induced by an irreducible sl (3)-module V_0, the structures of H~1(S(3,m), V), where V is an irreducible S(3,m)-module, and the structures of the restricted cohomology H_*~1(S(3,(1, 1, 1)), V), where V is an irreducible restricted S(3, (1, 1, 1))-module.     

ON DISTRIBUTIVE MODULES AND LOCALLY DISTRIBUTIVE RINGS

Distributive modules over artinian rings are characterized via module diagrams,andit is shown that a left artinian ring is(two-sided)locally distributive in case its leftindecomposable injective modules and projective modules are distributive.This latter resultis used to show that a locally distributive artinian ring and the endomorphism ring of itsminimal cogenerator have identical diagrams.     

On the Maximum Spectral Radius of （m, n）-trees with Given Diameter

Let (V, U) be the vertex-partition of tree T as a bipartite graph. T is called an (m,n)-tree if |V|=m and |U| = n. For given positive integers m,n and d, the maximum spectral radius of all (m,n)-trees on diameter d are obtained, and all extreme graphs are determined.     

关于$(m,n)$-凝聚模与预包络

In this paper, let m, n be two fixed positive integers and M be a right R-module, we define （m, n）-M-flat modules and （m, n）-coherent modules. A right R-module F is called （m, n）-M-flat if every homomorphism from an （n, m）-presented right R-module into F factors through a module in addM. A left S-module M is called an （m, n）-coherent module if MR is finitely presented, and for any （n, m）-presented right R-module K, Hom（K, M） is a finitely generated left S-module, where S = End（MR）. We mainly characterize （m, n）-coherent modules in terms of preenvelopes （which are monomorphism or epimorphism） of modules. Some properties of （m, n）-coherent rings and coherent rings are obtained as corollaries.     

DECOMPOSITIONS OF BOSONIC MODULES OF LIE ALGEBRAS W_(1 ∞) AND W_(1 ∞)(gl_N)

A bosonic construction (with central charge c = 2) of Lie algebras W1 ∞and W1 ∞(glN), as well as the decompositions into irreducible modules are described. And for Wi oo, when restricted to its Virasoro subalgebra Vir, a bosonic construction and the same decomposition for Vir are obtained.     

THE INVARIANT FORMS ON THE GRADED MODULES OF THE GRADED CARTAN TYPE LIE ALGEBRAS

This paper determines the graded modules of graded Cartan type Lie aigebras whichpossess nondegenerate invariant form.     

ON THE CONSTRUCTION OF INDECOMPOSABLE POSITIVE DEFINITE QUADRATIC FORMS OVER Z

The aim of this paper is to construct indecomposable positive definite Z-lattices withgiven rank n(2≤n≤9) and discriminant a,a being composite and square-free,but for afinite number of exceptions.And all exceptional cases for 3≤n≤9 are determined.Theseare unsettled cases of a paper of O’Meara in 1975.     

TORSION THEORIES AND PRIMARY DECOMPOSITION OF MODULES

In this paper,the author obtains the following result:If R is a left semi-artinian maxring,then the primary decomposition theorem holds for all R-modules if and only if Ext(S_1,S_2)=0 for any non-isomorphic simple modules S_1 and S_2.By mecns of the primarydecomposition of modules,the author proves:All torsion theories of ~Bμ are simple type ifand only if R isa left semi-artinian max ring and Ext(S_1,S_2)=0 for any non-isomorphicsimple modules S_1,S_2.     

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）.     

Strongly Irreducible Submodules of Modules

Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.     

Uniform Hypergraphs under Certain Intersection Constraints between Hyperedges

A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥ 0 are two integers.Let k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform if all its hyperedges have the same size k,and H is L-intersecting if the number of common vertices of every two hyperedges belongs to L.In this paper,we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting(n,m)-hypergraphs for fixed n,m ...     

New Simple Lie Algebras of Characteristic p

Over an algebraically closed field F of characteristic p>3, classes of Lie algebras \Sigma= \Sigma(n, m, r, G)(m>0), \Sigma^*=\Sigma^*(n,m, r,G)(n+2\ne 0(modp))and \tilde \Sigma=\tilde \Sigma(n,m, r, G)((n+2\equiv 0(modp))are constructed, where n, m are non-negative numbers, r = (s_0+1, s_1+1, \cdots, s_n+ 1, t_1+1,\cdots,t_n+l)is a(2n+l)-tuple of positive numbers and G is a subgroup of the additive group of F. \bar \Sigma,\Sigma^* and \tidle \Sigma are shown to be all simple Lie algebras with dimensions p^N - 2, p^N and p^N — 1 respectively, where $N=\sum\limits_i=0^n(s_i+1)+\sum\limits_i=1^n(t_i+1)+m$ Their derivation algebras are determined. It is shown that they are of generalized Cartam type K when m=0 and of generalized Cartan type H when m>0. It is then determined that \Sigma^* and \Sigma are new simple Lie algebras if n>0. Conditions of isomorphism are obtained. And a special graded algebra structure, the K-like gradation, is discussed.     

Indecomposable modules of the intermediate series over W(a, b)

For any complex parameters a and b,W(a,b)is the Lie algebra with basis{Li,Wi|i∈Z}and relations[Li,Lj]=(j i)Li+j,[Li,Wj]=(a+j+bi)Wi+j,[Wi,Wj]=0.In this paper,indecomposable modules of the intermediate series over W(a,b)are classified.It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module.Furthermore,if a∈/Q,an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wk’s.     

INTERSECTION OF PRIME SUBMODULES AND DIMENSION OF MODULES

The aim of this paper is to study the conditions by which a P-prime submodule can be expressed as a finite intersection or union of P-prime submodules. Also corresponding to dimension and rank of modules, some equivalent conditions for a ring to be a Dedekind domain are given.     

Monomial Hopf algebras over fields of positive characteristic

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n) are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.     

Irreducible weight modules over the twisted Schrödinger-Virasoro algebra

It is shown that the support of an irreducible weight module over the SchrSdinger-Virasoro Lie algebra with an infinite-dimensional weight space coincides with the weight lattice, and all nontrivial weight spaces of such a module are infinite-dimensional. As a by-product, it is obtained that every simple weight module over Lie algebra of this type with a nontrivial finite-dimensional weight space is a Harish-Chandra module.     

The lower and upper bounds of the degree distribution of a graded algebra

For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.     

Graphs of nonsolvable groups with four degree-vertices

Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.     

SELF-CANCELLATION OF MODULES HAVING THE FINITE EXCHANGE PROPERTY

Self-cancellation of modules having the finite exchange property is introduced. If a right R-module M has the finite exchange property, it is shown that M has self-cancellation if and only if EndR(M) is a strongly separative ring. Using this result, some new characterizations of strong separativity are obtained.