除環上一般線性群及么正群的結構重訪

The Structure of General Linear and Unitary Groups over Division Rings Revisited

林浚沂(Chun-Yi Lin)

指導教授 ： 余家富

國立臺灣大學/理學院/數學研究所/碩士(2022年)

https://doi.org/10.6342/NTU202202464

摘要

在這篇說明文中，我們重訪除環上一般線性群及么正群的結構，尤其特別關注這兩類群中某些子群的商群的單純性。

在第2章中，我們解釋了迪厄多內行列式的構造過程及除環上一般線性群的結構。我們主要的動機是去了解其中隱含的理論背景，及如何證明除環上射影特殊線性群的單純性。而後者也形成了一類重要的李型單純群。在這裡我們證明單純性的方式，是基於用來證明體上射影特殊線性群的單純性的岩澤理論。這樣的證明方式比埃米爾·阿廷在1957年出版的《幾何代數》一書中的證明容易。另外，我們也修正了在一些文章中，轉置矩陣在行列式上的關係。

在第3章中，我們的目標是闡明除環上么正群的結構。在他先前的著作中，讓·迪厄多內建立了除環上么正群的結構定理。本章的目的是對於迪厄多內的單純性定理，利用岩澤判據給出一個更有系統性的證明。為了達到這個目標，我們引進了艾克勒變換，他可以被視為是么正錯切的一個概念延伸。除此之外，為了解決某些特例，我們也解釋了么正錯切及特殊么正群的關係。

關鍵字

一般線性群 ； 迪厄多內行列式 ； 么正群 ； 岩澤判據 ； 迷向向量 ； 么正錯切

並列摘要

In this expository article, we revisit the structure of general linear and unitary groups over division rings, especially some subgroup quotient of these two types of groups.

In chapter 2, we explain the construction of the Dieudonné determinant and the structure of general linear groups over division rings. Our main motivation is to understand the underlying theoretic background and the proof of the simplicity of the projective special linear groups PSLn(K) over a division ring K. The latter gives an important family of simple groups of Lie type. The method of proving simplicity here is based on Iwasawa’s argument which proves the simplicity of PSLn(F), where F is a field. This is simpler than the proof given in E. Artin’s exposition [Geometric Algebra, Interscience Publishers, 1957]. Moreover, we also fix the relation on the determinants of the transposes of matrices in some literature.

In chapter 3, we aim to clarify the structure of unitary groups over division rings. In his pioneering works, Dieudonné established the structural theorems for
unitary groups over division rings. The purpose of this chapter is to give a more systematic proof of Dieudonné’s structural theorem using Iwasawa’s criterion. To achieve this goal, the Eichler transformations, which can be viewed as a promoted concept of unitary transvections, are introduced. In addition, we also explain the relation between unitary transvections and the special uniatry groups to solve some special cases.

並列關鍵字

general linear group ； Dieudonné determinant ； unitary group ； Iwasawa's criterion ； isotropic vector ； unitary transvection

參考文獻

[1] Semyon Alesker, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables. Bull. Sci. Math. 127 (2003), no. 1, 1–-35.

[2] Semyon Alesker, Non-commutative determinants and quaternionic Monge-Ampere equations. Advances in analysis and geometry, 289–300, Trends Math., Birkhauser, Basel, 2004.

[3] E. Artin, Geometric algebra. Interscience Publishers, Inc., New York-London, 1957.

[4] J. L. Brenner, Applications of the Dieudonné determinant. Linear Algebra Appl. 1 (1968), 511–536.

[5] Peter J. Cameron, Notes on Classical Groups. Available in https://webspace.maths.qmul.ac.uk/p.j.cameron/class_gps/cg.pdf