Kac-Moody代数的实根向量和虚根向量及Schubert-子模的一些性质
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摘要
本文分为两个相对独立的篇章：
第一部分主要讨论了Kac-Moody代数中的一类基本问题，即给定一个实根或虚根，其对应的实根向量和虚根向量该如何表示?我们要求给定的广义Cartan矩阵满足条件或，对；对应的李代数记为g(A)。我们得到结果如下：[定理1]设广义Cartan矩阵满足以下条件：
(ⅰ)1≤i≠j≤n时，a_(i,j)=0或a_(i,j)＜-2；
(ⅱ)Kac-moody代数g(A)的Weyl群W中元素w_j的简约表示为：
(ⅲ)任何两个相邻的因子r_(ik)和r_(ik-1)不可交换。记实根其相应的实根向量为：则有：[定理2]条件及符号如同定理1中所述，设t_(j-1)满足条件
贝：
／1＼一＜／jj一1，O；＞＋ti＿1（Qi。j。。十工），。＼一＜U。＿，、口y＞一t。1t，＿
在第二部分，我们主要关注Schubert－子模的维数问题。M．E．Hall
在他的博士论文中，已经讨论过Schubert－伸和可积最高权模之间所
具有的紧密联系，包括可积最高权模可表成Schubert－子模的并；且尽
管可积最高权模的维数本身是无限的，但Schubert－子模的维数总是有
限的；等等。本文中，我们试图对SChllbeyt－子模的维数作些探讨。
本部分组织如下：
第一节是引言，主要给出Schubert－子模的定义以及M．E．Hall在他
的论文中已经得到的结果。
第二节是概念和背景知识，主要是给出一些本文用到的概念及符
号，以及一些本文用到的与W的性质相关的引理；并着重介绍引理2．2，
弓理2．3，弓理2．4。
第三节是一些既有的结果。从这一节起。我们只讨论对应于广义
／2－n＼
Cartan矩阵A＝DIn＞2的一类特殊的二阶双曲型李代数。
＼2－n j
特别介绍引理s2中关于g卜）的实根集的描述。
在第四节中，令咤表示某些根空间的和。然后我们证明咤是交
2
换李代数，并且 Schubert－子模儿＝U（n￡）．vw（。）。
第五节我们先证明了g卜）的实根和虚根的几个性质，然后对
Schubert－子模的维数表示给出猜想，这一猜想的证明在目前的证明
中尚有困难。
There are two parts in this article.
Part I is mainly discussing an elementary problem in Kac-Moody Algebra:how to describe the real and imaginary root vectors corresponding to a given real or imaginary root? We reqire the generalized Cartan matrice satisfy the following condition that a- is zero or less than -2 for all 1 i,j (iii)for all k{2,3,...}, r and r are not exchangeable. Let denote the real roots: Let X. denote the corresponding real root vectors: Then:
[Theorem 2] The conditions same as [Theorem 1]. Suppose that tj-1
satisfies:Then,
In Part II,we are specifically interested at dimensions of Schubert submodules. We got to know the close relationships between an integrable highest-weight module and Schubert submodules in the paper by M.E.Hall. He demenstrates some properties of Schubert submodules including the fact that an integrable highest-weight module V is a union of it's Schubert submodules and the dimensions of Schubert submodules are finite although the integrable highest-weight module itself is infinite-dimensional. It's the purpose to find an expression to dimensions of Schubert submodules in this part.
This part is oganized as follows:
Chapter 1 is an introduction. We present the definition of Schubert submodules and what M.E.Hall has done.
Chapter 2 is preliminaries. We cite some relevant lemmas to introduce the properties of w,which is an element in Weyl group of a Lie algebra.
Chapter 3 is some results associated with a special kind of Lie al-
gebras. These Lie algebras are hyperbolic corresponding to a generalized
Cartan matrice A =
n>2. We specially introduce lemma
3.3,an expression of the real set of g(A).
Then in chapter 4,we denote n as the sum of some root spaces ga. And we prove that n is a commutative Lie algebra and a Schubert submodule Vw can be expressed by Vw = U(n,}.vw.
At last in chapter 5,we give proofs to some properties of real and imaginary roots of g(A) and a guess to express the dimensions of Schubert submodules,whose proof is encountering some difficulty at present.
第一部分主要讨论了Kac-Moody代数中的一类基本问题，即给定一个实根或虚根，其对应的实根向量和虚根向量该如何表示?我们要求给定的广义Cartan矩阵满足条件或，对；对应的李代数记为g(A)。我们得到结果如下：[定理1]设广义Cartan矩阵满足以下条件：
(ⅰ)1≤i≠j≤n时，a_(i,j)=0或a_(i,j)＜-2；
(ⅱ)Kac-moody代数g(A)的Weyl群W中元素w_j的简约表示为：
(ⅲ)任何两个相邻的因子r_(ik)和r_(ik-1)不可交换。记实根其相应的实根向量为：则有：[定理2]条件及符号如同定理1中所述，设t_(j-1)满足条件
贝：
／1＼一＜／jj一1，O；＞＋ti＿1（Qi。j。。十工），。＼一＜U。＿，、口y＞一t。1t，＿
在第二部分，我们主要关注Schubert－子模的维数问题。M．E．Hall
在他的博士论文中，已经讨论过Schubert－伸和可积最高权模之间所
具有的紧密联系，包括可积最高权模可表成Schubert－子模的并；且尽
管可积最高权模的维数本身是无限的，但Schubert－子模的维数总是有
限的；等等。本文中，我们试图对SChllbeyt－子模的维数作些探讨。
本部分组织如下：
第一节是引言，主要给出Schubert－子模的定义以及M．E．Hall在他
的论文中已经得到的结果。
第二节是概念和背景知识，主要是给出一些本文用到的概念及符
号，以及一些本文用到的与W的性质相关的引理；并着重介绍引理2．2，
弓理2．3，弓理2．4。
第三节是一些既有的结果。从这一节起。我们只讨论对应于广义
／2－n＼
Cartan矩阵A＝DIn＞2的一类特殊的二阶双曲型李代数。
＼2－n j
特别介绍引理s2中关于g卜）的实根集的描述。
在第四节中，令咤表示某些根空间的和。然后我们证明咤是交
2
换李代数，并且 Schubert－子模儿＝U（n￡）．vw（。）。
第五节我们先证明了g卜）的实根和虚根的几个性质，然后对
Schubert－子模的维数表示给出猜想，这一猜想的证明在目前的证明
中尚有困难。
There are two parts in this article.
Part I is mainly discussing an elementary problem in Kac-Moody Algebra:how to describe the real and imaginary root vectors corresponding to a given real or imaginary root? We reqire the generalized Cartan matrice satisfy the following condition that a- is zero or less than -2 for all 1 i,j
[Theorem 2] The conditions same as [Theorem 1]. Suppose that tj-1
satisfies:Then,
In Part II,we are specifically interested at dimensions of Schubert submodules. We got to know the close relationships between an integrable highest-weight module and Schubert submodules in the paper by M.E.Hall. He demenstrates some properties of Schubert submodules including the fact that an integrable highest-weight module V is a union of it's Schubert submodules and the dimensions of Schubert submodules are finite although the integrable highest-weight module itself is infinite-dimensional. It's the purpose to find an expression to dimensions of Schubert submodules in this part.
This part is oganized as follows:
Chapter 1 is an introduction. We present the definition of Schubert submodules and what M.E.Hall has done.
Chapter 2 is preliminaries. We cite some relevant lemmas to introduce the properties of w,which is an element in Weyl group of a Lie algebra.
Chapter 3 is some results associated with a special kind of Lie al-
gebras. These Lie algebras are hyperbolic corresponding to a generalized
Cartan matrice A =
n>2. We specially introduce lemma
3.3,an expression of the real set of g(A).
Then in chapter 4,we denote n as the sum of some root spaces ga. And we prove that n is a commutative Lie algebra and a Schubert submodule Vw can be expressed by Vw = U(n,}.vw.
At last in chapter 5,we give proofs to some properties of real and imaginary roots of g(A) and a guess to express the dimensions of Schubert submodules,whose proof is encountering some difficulty at present.
引文
1 M.E.Hall:An Introduction to Schubert Submodules(Doctorial dissertation, not found in published magzines)
2 V.G.Kac: Infinite Dimensinal Lie Algebra, Third Edition, Cambridge University Press, Cambridge, 1990.
3 卢才辉、王秋媛：秩为2的对称双曲型的正实根集与广义Fibonacci数列的关系(王秋媛毕业论文，未发表)
4 万哲先：Kac-Moody代数导引，科学出版社，1993年。
5 孟道骥：复半单李代数引论，北京大学出版社
2 V.G.Kac: Infinite Dimensinal Lie Algebra, Third Edition, Cambridge University Press, Cambridge, 1990.
3 卢才辉、王秋媛：秩为2的对称双曲型的正实根集与广义Fibonacci数列的关系(王秋媛毕业论文，未发表)
4 万哲先：Kac-Moody代数导引，科学出版社，1993年。
5 孟道骥：复半单李代数引论，北京大学出版社