素特征的"李定理"

设L为代数闭域F上有限维李代数,著名的李定理说若char F=0,则L为可解当且仅当L的任一有限维不可约模为1维的.在这里特征为0及模为有限维两个条件都是本质的.(1)若charF=p＞o,则L为交换当且仅当L的任一(有限维)不可约模为1维的;(2)若char F=0,则L为交换当且仅当L的任一(有限维或无限维)不可约模为1维的;(3)若charF=p＞7,L为李代数(限制李代数),则L为可解当且仅当L的任一不可约模(限制模)的维数为p的幂.     

阶化Cartan型代数S(m;n)的不可约表示及其约化

设L=S(m;n)是定义在特征P＞3的代数闭合域F上的阶化特殊型李代数,利用已研究L的不可约表示的方法,通过定义L的如下阶化:限制情形定义L=(田)q≥-1 L[q],I,非限制情形定义(L)=(田)q≥-1 (L)[q],I,这里是L的本原P-包络,有表达式(L)=(田)mΣi=1 ni-1Σdi=1 FDpidi,而I是{1,2,…,m)的子集,得到当P-特征标x是正则半单时,在限制李代数情形所有不可约Ux(L)-模都是从不可约Ux(L[0],I)-模诱导的;在非限制的情形,所有不可约U(x)(L),(Upx(L,x))-模都是从不可约L(x)_(L[0],I)-模诱导的,这里(x)是x到(L)*上的平凡扩张.     

具有三角分解可解李代数的表示

本文研究具有三角分解可解李代数和它的表示,探讨了具有三角分解可解李代数是广义限制李代数的条件,对于某些s∈Map(B,F),在uφ2(L,S)-模的范畴里,确定了不可约模和主不可分解模,并对upuφ2,L,S)的块进行了描述.     

单的n+1维n-李代数的表示

本文证明了不可约的L(A)-模是A-模的充要条件,给出了单的n 1-维n-李代数的有限维不可约表示的分类．     

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.     

具有β(L)=m-3的m-维非交换3-李代数的分类*

对特征零域F上具有β(L)=m-3的m-维非交换3-李代数的结构进行了分类, 且给出了每一类3-李代数的具体乘法结构.证明了满足β(L)=m-3且中心含在导代数的非交换3-李代数的维数小于等于11,大于等于5, 且导代数维数等于1时的3-李代数仅有两类, 导代数维数等于2 时有24类.     

单的n+1维n-李代数的表示(英)

本文证明了不可约的$L(A)$-模是$A$-模的充要条件,给出了单的$n+1$-维$n$-李代数的有限维不可约表示的分类.     

最简线状n-李代数

定义了一类特殊的幂零n-李代数,即最简线状n-李代数,它是最简线状李代数的推广.确定了m维最简线状n-李代数A的导子代数Der(A)和自同构群Aut(A),定义了n-李代数的全形h(A)=Der(A)( ) A,并证明了当A的基域F的特征P为零或P＞m-n时,Der(A)是不可解的完备李代数,而h(A)的一个子代数是可解的完备李代数,当F的特征为零时,Aut(A)是无中心的不可解群.     

带三角分解李代数的赋值模

设F是特征零的域，L是F上的带三角分解的李代数，L^-是相应的Loop代数．本文将定义L^-上赋值模的概念，并给出其不可约模的张量积是不可约模的等价条件．     

Jacobson猜想成立的某些充分条件

Jacobson在文献[1]给出了一个猜想:若(L,[p])为限制李代数,且x^[p]n(x)=x,?x∈L,n(x)>0,则L是交换李代数.至今为止,人们还不知此猜想是否正确.本文分别证明这个猜想在p映射为p半线性映射或者域F为代数闭域条件下的正确性.     

RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p ^MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.     

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.     

Gröbner-Shirshov Bases of Irreducible Modules of the Quantum Group of Type G2

First, the authors give a Grbner-Shirshov basis of the finite-dimensional irreducible module Vq(λ) of the Drinfeld-Jimbo quantum group U_q(G_2) by using the double free module method and the known Grbner-Shirshov basis of U_q(G_2). Then, by specializing a suitable version of U_q(G_2) at q = 1, they get a Grbner-Shirshov basis of the universal enveloping algebra U(G_2) of the simple Lie algebra of type G_2 and the finite-dimensional irreducible U(G_2)-module V(λ).     

Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions.     

Some Theorems on Leibniz Algebras

If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.     

The Lie Algebras in which Every Subspace s Its Subalgebra

In this paper, we study the Lie algebras in which every subspace is its subalgebra (denoted by HB Lie algebras). We get that a nonabelian Lie algebra is an HB Lie algebra if and only if it is isomorphic to g+Cidg, where g is an abelian Lie algebra. Moreover we show that the derivation algebra and the holomorph of a nonabelian HB Lie algebra are complete.     

The Lie Algebras in which Every Subspace Is Its Subalgebra

In this paper, we study the Lie algebras in which every subspace is its subalgebra (denoted by HB Lie algebras). We get that a nonabelian Lie algebra is an HB Lie algebra if and only if it is isomorphic to $g\dot{+}\mathbb{C}id_g$, where $g$ is an abelian Lie algebra. Moreover we show that the derivation algebra and the holomorph of a nonabelian HB Lie algebra are complete.     

Enveloping algebras that are principal ideal rings

Let L be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of L is a principal ideal ring if and only if L is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.     

A noncommutative Brooks-Jewett Theorem

In classical measure theory the Brooks-Jewett Theorem provides a finitely-additive-analogue to the Vitali-Hahn-Saks Theorem. In this paper, it is studied whether the Brooks-Jewett Theorem allows for a noncommutative extension. It will be seen that, in general, a bona-fide extension is not valid. Indeed, it will be shown that a C^*-algebra A satisfies the Brooks-Jewett property if, and only if, it is Grothendieck, and every irreducible representation of A is finite-dimensional; and a von Neumann algebra satisfies the Brooks-Jewett property if, and only if, it is topologically equivalent to an abelian algebra.