扩张Hamilton李超代数的Borel子代数

设为特征零的代数闭域上秩为5的有限维Z-分次Hamilton单李超代数H通过添加次数导子得到的扩张李超代数.本文通过对正则元的分类,证明了关于典范环面共有160个正根系,从而得到160个Borel子代数;通过单根以及连接的定义,确定了每一个正根系的单根系,进而刻画了任意两个Borel子代数的连接关系;最后证明了共有48个Borel子代数是极大可解子代数.本文所得结果可用于进一步研究Cartan型单李超代数的结构与表示.     

特殊Cartan型李超代数的Borel子代数

本文研究了特征零的代数闭域上秩为4的有限维特殊Cartan型李超代数S的结构.利用正则元的划分,确定出S关于典范环面的所有正根系,从而得到了S的所有Borel子代数;对于每一个正根系,通过给出其单根系,得到了任何两个Borel子代数的连接关系;最后确定了每一个Borel子代数的极大可解性.本文所得结果可用于进一步研究Cartan型单李超代数的结构与表示.     

任意随机变量序列泛函的强极限定理(Ⅱ)

当{Xn，n≥0}是任意随机变量序列，fn(x0，…，xn)(n≥1)是R^n 1上有界Borel可测函数时，利用鞅的理论，建立了泛函序列{fn(X0，…，Xn)，n≥1)的强极限定理，作为推论，得出了关于任意随机变量序列到达某些Borel可测集的频率与条件概率之间的关系．     

交换环上严格上三角矩阵的广义李三导子(英文)

本文研究了交换环R上所有n×n严格上三角矩阵构成的李代数N(n,R)(n≥5)上广义李三导子.利用矩阵技巧,证明了N(n,R)(n≥5)上任意广义李三导子为一李三导子与一位似映射的和.对于N(n,R)(n≥3)上广义李导子,得出类似结果.     

全相似保序变换半群H(SPO_n,r)的秩

设自然数n≥5,X_n={1,2,…,n},并赋予自然数的大小顺序,令H(SPO_n,r)=SPO_n∪L(n,r)(5≤n,2≤r≤n-3)是X_n上的全相似部分保序变换半群.得到全相似保序变换半群H(SPO_n,r)的秩为■.     

Cartan型李代数W(n;m)的一类Borel子代数

构造了Cartan型李代数W(n;m)的一类Borel子代数φ(n;m),其中n是一个正整数,且m=(m_1,…,m_n)是一个n-元正整数数组.确定了φ(n;m)的导子代数.特别地,φ(n;1)是一个Cartan型完备阶化李代数,它不同于任何典型完备李代数.     

正合Borel子代数的旗导出的强共轭作用

设拟遗传代数A有强正合Borel子代数. 证明了对$A$的任意标准半单子代数S,均存在A的正合Borel子代数B, 使得S是B的极大半单子代数. 由正合Borel子代数构成的旗的最大长度等于A的半径与其Grothendick群的秩之差加1. 正合Borel子代数的共轭类惟一当且仅当A是basic代数;正合Borel子代数的共轭类个数为2当且仅当A是半单代数.非basic非半单拟遗传代数的正合Borel子代数的共轭类个数或者为0或者不小于3. 正合Borel子代数的内自同构群诱导出在A上的强共轭作用, 其全体的集合与$A$的全体极大半单子代数之集合的强轨道的集合一一对应. 所有正合Borel子代数彼此是基共轭的, 即若B与C均是A的正合Borel子代数, 则存在A的幂等元e及可逆元u, 使得eCe=u^-1eBeu.     

Full环上O_n~ (V)的自同构

局部环上正交群O_n(V)(n≥5,v≥1,2,3,5为单位)的自同构由B,R.McDonald定出作者曾与王仁发定出半局部环上O_n(V)(n≥5,v≥1,2,3,5为单位)的自同构。H.Ishibashi讨论了full环上O_n(V)的结构。本文在此基础上利用局部化方法定出了full环上O_n~ (V)(n≥7,v≥1,2,3,5为单位)的自同构,其形式为∧=P_xcφ_g,这里g是一保持正交性的半线性同构,P_a为放射自同构。     

拟遗传代数的特征模与张量积

本文研究零关系拟遗传代数A的模范畴与其正合Borel子代数B的模范畴之间的联系.证明了正合Borel子代数B的诱导模范畴完全含于A的好模范畴, 即好模诱导好模. 特别, 设A是具有纯粹强正合子代数B的零关系拟遗传代数,则A的特征模是B的特征模通过正合函子-_BA的诱导模当且仅当内射B-模的诱导A-模作为B-模仍是内射的. 本文还证明了基拟遗传单列代数的正合Borel子代数是右单列的, 并且其特征模恰是它的正合Borel子代数的特征模的诱导模.     

拟遗传代数的诱导模与广义Betti数

本文引入了模的广义Betti数,给出了经典Betti数与广义Betti数之间的关系,证明了具有纯粹强正合Borel子代数的零关系拟遗传代数的诱导模的极小投射分解可通过其正合Borel子代数的相应模的极小投射分解的诱导给出,从而两者具有相同的广义Betti数.     

Ising vectors and automorphism groups of commutant subalgebras related to root systems

In this article we study and obtain a classification of Ising vectors in vertex operator algebras (VOAs) associated to binary codes and times root lattices, where an Ising vector is a conformal vector with central charge 1/2 generating a simple Virasoro sub-VOA. Then we apply our results to study certain commutant subalgebras related to root systems. We completely classify all Ising vectors in such commutant subalgebras and determine their full automorphism groups. C. H. Lam was supported by NSC grant 93-2115-M-006-012 of Taiwan and National Center for Theoretical Sciences, Taiwan, S. Sakuma was supported by JSPS Research Fellowships for Young Scientists and H. Yamauchi was supported by JSPS Research Fellowships for Young Scientists     

Affine Slice for the Coadjoint Action of a Class of Biparabolic Subalgebras of a Semisimple Lie Algebra

In this article, we give a simple explicit construction of an affine slice for the coadjoint action of a certain class of biparabolic (also called seaweed) subalgebras of a semisimple Lie algebra over an algebraically closed field of characteristic zero. In particular, this class includes all Borel subalgebras.     

Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko.     

Strong conjugation actions induced by flags of exact Borel subalgebras

Let A be a quasi-hereditary algebra with a strong exact Borel subalgebra. It is proved that for any standard semisimple subalgebra T there exists an exact Borel subalgebra B of A such that T is a maximal semisimple subalgebra of B. It is shown that the maximal length of flags of exact Borel subalgebras of A is the difference of the radium and the rank of Grothendic group of A plus 2. The number of conjugation-classes of exact Borel subalgebras is 1 if and only if A is basic; the number is 2 if and only if A is semisimple. For all other cases, this number is 0 or no less than 3. Furthermore, it is shown that all the exact Borel subalgebras are idempotent-conjugate to each other, that is, for any exact Borel subalgebras B and C of A, there exists an idempotent e of A, and an invertible element u of A, such that eBe = u^-1eCeu.     

Ad-nilpotent ideals of a parabolic subalgebra

We extend the results of Cellini and Papi [P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra, J. Algebra 225 (2000) 130–140; P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra II, J. Algebra 258 (2002) 112–121] on the characterizations of ad-nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical simple Lie algebra, we give formulas for the number of these ideals.     

非扭仿射Kac-Moody代数中Borel子代数的扩代数

决定了非扭仿射Kac-Moody代数中所有包含标准Borel子代数的子代数。     

Borel Subalgebras of Schur Superalgebras

It is proved that any Schur superalgebra is representable as a product of two Borel subalgebras of that superalgebra, which are symmetric w.r.t. its natural anti-isomorphism (Bruhat-Tits decomposition). This readily implies that any simple module is uniquely defined by its highest weight, and all other weights are strictly less than is the highest under the dominant ordering. It is stated that the fundamental theorem of Kempf, which is valid for all classical Schur algebras, might be true for superalgebras only if they are semisimple. Nevertheless, a weaker theorem of Grothendieck holds true for superalgebras since Borel subalgebras are quasihereditary. Also we formulate an analog of the Donkin-Mathieu theorem for Schur superalgebras, and show that it is valid in the elementary non-classical case, that is, for the algebras S(1|1, r).__________Translated from Algebra i Logika, Vol. 44, No. 3, pp. 305–334, May–June, 2005.