Restricted Representations of Lie Superalgebras of Hamiltonian Type

Let F be an algebraically closed field of prime characteristic p?>?2, H(n) the Hamiltonian Lie superalgebra over F. The simple restricted H(n)-modules are studied and classified. Furthermore, a sufficient and necessary condition is provided for restricted Kac modules to be simple.     

局限李超代数的局限表示

本文对如下结果给出一个构造性、初等的、简洁的证明:有限维局限李超代数的有限维局限表示均完全可约当且仅当它的奇部为0而偶部是一个环面.这使得李代数中相应结果成为特例.     

On Blocks in Restricted Representations of Lie Superalgebras of Cartan Type

Let g be a restricted Lie superalgebra of Cartan type W (n), S(n) or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac's definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero.     

Vertex Representations of Toroidal Special Linear Lie Superalgebras

Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, a free field representation of toroidal Lie superalgebras of type A(m, n) is constructed using both vertex operators and bosonic fields.     

Cartan型广义李超代数

设F是特征不为2的域.本文定义了F上的广义李超代数，证明了Z-阶化广义李超代数的单性准则.然后定义了有限维Cartan型广义李超代数W(n),证明了W(n)的单性.最后指出对Cartan型广义李超代数S(n)与H(n)，亦有与W(n)相似的结果.     

Superderivations for a Family of Lie Superalgebras of Special Type

By means of generators, superderivations are completely determined for a family of Lie superalgebras of special type, the tensor products of the exterior algebras and the finite-dimensional special Lie algebras over a field of characteristic p〉3. In particular, the structure of the outer superderivation algebra is concretely formulated and the dimension of the first cohomology group is given.     

奇Hamiltonian型滤过李超代数HO

本文应用象空间极小维数的方法研究了Cartan型模李超代数HO的非平凡可迁滤过(HO,R)的不变性.应用滤过的不变性得到模李超代数HO的某些内蕴性质,从而得到此类李超代数在同构意义下的分类.     

Howe Duality for Lie Superalgebras

We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest-weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition into irreducible gl(m|n)-modules of the symmetric and skew-symmetric algebras of the symmetric square of the natural representation of gl(m|n). In the former case, we also find explicit formulas for the highest-weight vectors. Our work unifies and generalizes the classical results in symmetric and skew-symmetric models and admits several applications.     

一类广义W型李超代数

构造了一类有限维广义Cartan型模李超代数W,并证明了它是李超代数W(n)的一个扩张,进而决定了它的导子超代数.     

阶化 Cartan 型李超代数的结合型

本文确定了具有非退化结合型的Ｃａｒｔａｎ型李超代数。     

有限维Cartan型模李超代数的保积Hom-结构

在特征大于3的代数闭域上,利用Cartan型模李超代数的自然滤过不变性,通过计算Cartan型模李超代数的Hom-映射在其-1分支上的作用,本文证明了有限维Cartan型模李超代数的保积Hom-结构是平凡的.     

非单W和S型李超代数的滤过和分类

设F是特征p3的域,g是F上外代数与有限维广义Witt李代数或特殊李代数的张量积所构成的有限维非单李超代数.本文通过确定ad幂零元的方法证明了g的标准滤过是g的自同构群的不变量,进而证明了定义这类李超代数的参数组是内蕴的,从而给出了这两类李超代数在同构意义下的分类.     

Irreducible Quasi-Finite Representations of a Block Type Lie Algebra

Let B be the Block type Lie algebra over ? with basis {L _{α, i} , C ₁, C ₂ | (α, i) ∈ ? × ? \ {(0, ? 2)}} and Lie bracket [L _{α, i} , L _{β, j} ] = (β(i + 1) ? α(j + 1))L _{α+β, i+j}  + αδ_{α+β, 0}δ _{i+j, ?2} C ₁ + (i + 1)δ_{α+β, 0}δ _{i+j, ?2} C ₂, where C ₁, C ₂ are central elements. In this paper, it is proved that a quasi-finite irreducible B-module is either a highest or a lowest weight module. We also give a classification of all highest/lowest weight B-modules.     

系数在结合超代数的矩阵李超代数的2-上循环

设α是域F上的结合超代数满足[α,α]=α或α=F．证明了当m n>1时,H2(glm|n(α),F)(?)HC1(α,F)．定义了一大类广义微分算子李超代数,作为W-无穷代数W∞(glN)的推广．确定了这些李超代数的2-上循环．同时给出了矩阵量子微分算子李超代数的2-上同调群．     

限制Hamiltonian型及Contact型李代数的不可约表示

设L=H（2r;1）或K（2r+1;1）是定义在特征p>2的代数封闭域F上的限制Hamiltonian型或Contact型李代数.在对广义Jacobson-Witt代数及特殊代数不可约表示的研究基础上,通过定义L的如下阶化:L=L_[q],I,其中I是{1,2,…,r}的子集,得到当p-特征函数χ是正则半单时,所有不可约U_χ（L）-模都是从不可约U_χ(L_[O].I)-模诱导的.     

李超代数的泛包络代数同构问题

李超代数的一个性质P叫做关于泛包络代数的不变量,如果对于任意李超代数L,H,只要L具有性质P,并且泛包络代数U(L)和U(H)作为结合超代数是同构的,那么H亦具有性质P.通过讨论李超代数关于泛包络代数的不变量证明了:如果L的幂零长度不超过2,那么L和H是同构的.     

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra ■(m; 1) ■∧(n), where ■(m; 1) is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type...     

Mod p Reducibility of Unramified Representations of Finite Groups of Lie Type

Dedicated to the memory of Professor A. I. Kostrikin The main problem under discussion is to determine, for quasi-simplegroups of Lie type G, irreducible representations of G thatremain irreducible under reduction modulo the natural primep. The method is new. It works only for p >3 and for representations that can be realized over an unramified extension of Q_p, thefield of p -adic numbers. Under these assumptions, the mainresult says that the trivial and the Steinberg representationsof G are the only representations in question provided G isnot of type A1. This is not true for G=SL(2, p). The paper containsa result of independent interest on infinitesimally irrreduciblerepresentations of G over an algebraically closed field ofcharacteristic p. Assuming that G is not of rank 1 and G G₂(5),it is proved that either the Jordan normal form of a root elementcontains a block of size d with 1<d<p -1 or the highestweight of is equal to p -1 times the sum of the fundamentalweights. 2000 Mathematical Subject Classification: 20C33, 20G15.     

交换环上某些线性李超代数的理想

设Ｒ 是有１的交换环，２是Ｒ 的单位．设Ｌ 为环Ｒ 上的特殊线性李超代数或正交 辛李超代数．讨论了Ｌ 的理想与Ｒ 的理想的关系，证明了Ｌ 的所有理想都是标准的．