有限域上型Bn的Chevalley群之间的同态

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摘要

典型群是由李代数经过适当的运算构成的.本文所要讨论的 Chevalley 群就是

法国数学家 C.Chevalley 仿照 Lie 群的构造法,后又被 Steinberg、B.Kostant 等人推

广得来的.根据素根系的 Dynkin 图,有限维单李代数又分为

, 1;

A n 

, 2;

B n 

3;n

, 4

D n 

四大典型李代数和

6 7 8 4 2

, , , ,E E E F G

五个例外李代数.对应于典型群也

有型

, 1;

A n 

, 2;

B n 

, 3; , 4

n n

C n D n 

;

6 7 8 4 2

, , , ,E E E F G

九大类.

典型群的同态问题始终是典型群理论的中心问题之一.本文主要研究特征为

的有限域上型

的Chevalley 群的同态问题.

本文首先研究了特征为

的有限域上型

的Chevalley 群的结构和它的抛物

子群的性质.

其次,利用归纳法研究了特征为

的有限域上型

的Chevalley 群的非平凡同

态.设



为型

的不可约根系



的最高根,

{ | ( , ) 0, }I

   

  

为



的子根系.

当

2n

时,

为型

的根系;当

3n

时,

为一个型

1 1

A A

的根系; 当

3n

时,

为一个型

1 2n

A B 



的根系.利用归纳法,本文证明了主要结果

设



是型

的不可约根系,

是任意特征为

且

2p

的有限域,对任意

非平凡同态

: ( , , ) ( , , )G f G F





    

,则存在一个由

到

的域同态诱导的群同

态



和由

( )u U F





诱导的内自同构



和一个对角自同构



使得

( )

ux x



  

 

( , , )x G f   

关键词：有限域 Chevalley 群同态归纳法

ABSTRACT

Classical groups was constituted by the appropriate operation of Lie Algebras.

Chevvalley group which this paper will determine was imitated of the construction of

Lie group by the French mathematician C.Chevalley, and was expanded by Steinberg,

Kostant and others afterwords. According to the Dynkin diagrams of irreducible root

systems, finite dimensional simple Lie algebras were divided into four classical algebras

of types

, 1;

A n 

, 2;

B n 

C n

3

;

, 4

D n 

and five exceptional algebras of

types

6 7 8

, ,E E E

4 2

, ,F G

. Correspond to finite dimensional simple Lie algebras, there

exist nine types of Chevalley groups such as

, 1;

A n 

, 2;

B n 

, 3; , 4

n n

C n D n 

;

7 8

,E E

4,F

The problem of homomorphisms between classical groups is always one of focal

point in classical groups theory. In the present paper,we are going to study the

homomorphisms between Chevalley groups of type

over finite fields of characteristic

In the paper, we first discuss the construction of Chavelley groups of type

over

finite fields of characteristic

and the properties of the parabolic subgroups.

Secondly, we make use of induction to investigate the nontrivial homomorphisms

of Chevalley groups of type

over finite fields of characteristic

. Let



be a unique

highest root of the indecomposable root system



of type

be the subset of



including of all simple roots



with

( , ) 0

 



.If

2n

is the root system of type

; If

3n

is the root system of type of

1 1

A A

; If

3n

is the root system of

type of

A

B

. We will verify the main theorem by induction:

Let



be the indecomposable root system of type of

and

be any finite

fields of characteristic

, and

2p

. For any nontrivial homomorphsism

: ( , , ) (G f G



   

, , )F



there exists a field homomorphism



from

and an inner automorphism



induced

( )u U F





and a diagonal automorphism



( , , )G F 

 

such that

( )

ux x



  

 

for any

( , , )x G f  

Key words: Finite fields , Chevalley groups, Homomorphisms,

Induction

中文摘要

ABSTRACT

第一章绪论..........................................................1

§1.1 选题背景.................................................1

§1.2 预备知识.................................................2

第二章 Chevalley 群的结构............................................10

§2.1 Chevalley 群.............................................10

§2.2 Bruhat 分解和 Birkhoff 分解...............................11

§2.3 Chevalley 群的 Levi 分解..................................12

§2.4 Chevalley 群的自同构.....................................13

第三章有限域上型

的 Chevalley 群的同态............................16

§3.1



的规范化..............................................16

§3.2 可归纳假设的条件........................................19

§3.3 主要结果的证明..........................................19

第四章总结.........................................................21

参考文献............................................................22

在读期间公开发表的论文和承担科研项目及取得成果......................24

致谢................................................................25

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摘要：

摘要典型群是由李代数经过适当的运算构成的.本文所要讨论的Chevalley群就是法国数学家C.Chevalley仿照Lie群的构造法,后又被Steinberg、B.Kostant等人推广得来的.根据素根系的Dynkin图,有限维单李代数又分为,1;nAn,2;nBn,nC3;n,4nDn四大典型李代数和67842,,,,EEEFG五个例外李代数.对应于典型群也有型,1;nAn,2;nBn,3;,4nnCnDn;67842,,,,EEEFG九大类.典型群的同态问题始终是典型群理论的中心问题之一.本文主要研究特征为p的有限域上型nB的Chevalley群的同态问题.本文首先研究了特...

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