代數(shù)幾何IV

前言

　　要使我國的數(shù)學(xué)事業(yè)更好地發(fā)展起來，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強(qiáng)對數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強(qiáng)數(shù)學(xué)的出版工作?！　某霭娣矫鎭碇v，除了較好較快地出版我們自己的成果外，引進(jìn)國外的先進(jìn)出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社?？茖W(xué)出版社影印一批他們出版的好的新書，使我國廣大數(shù)學(xué)家能以較低的價格購買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無疑是對推動我國數(shù)學(xué)的科研與教學(xué)十分有益的事?！　∵@次科學(xué)出版社購買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計(jì)算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計(jì)16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙?，都是由該領(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊。對從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點(diǎn)，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計(jì)算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學(xué)獎”。這些大數(shù)學(xué)家的著作無疑將會對我國的科研人員起到非常好的指導(dǎo)作用?！　‘?dāng)然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項(xiàng)工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?！　】傊覍茖W(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內(nèi)容概要

This book contains two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory. The first part is written by T. A. Springer, a well-known expert in the first mentioned field. Hc presents a comprehensive survey, which contains numerous sketched proofs and he discusses the particular features of algebraic groups over special fields (finite, local, and global). The authors of part two-E. B. Vinbcrg and V. L. Popov-arc among the most active researchers in invariant theory. The last 20 years have bccn a period of vigorous development in this field duc to the influence of modern methods from algebraic geometry. The book will bc very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

書籍目錄

I.Linear algebraic Groups  Introduction  Historical Comments  Chapter 1.Linear Algebraic Groups over an Algebraically    1.Recollections from Algebraic Geometry     1.1.Affine Varieties     1.2.Morphisms     1.3.Some Topological Properties     1.4.Tangent Spaces     1.5.Properties of Morphisms     1.6.Non-Affine Varieties    2.Linear Algebraic Groups, Basic Definitions and Properties     2.1.The Definition of a Linear Algebraic Group     2.2.Some Basic Facts     2.3.G-Spaces     2.4.The Lie Algebra of an Algebraic Group     2.5.Quotients    3.Structural Properties of Linear Algebraic Groups     3.1.Jordan Decomposition and Related Results     3.2.Diagonalizable Groups and Tori     3.3.One-Dimensional Connected Groups     3.4.Connected Solvable Groups     3.5.Parabolic Subgroups and Borel Subgroups     3.6.Radicals, Semi-simple and Reductive Groups    4.Reductive Groups     4.1.Groups of Rank One     4.2.The Root Datum and the Root System     4.3.Basic Properties of Reductive Groups     4.4.Existence and Uniqueness Theorems for Reductive Groups     4.5.Classification of Quasi-simple Linear Algebraic Groups     4.6.Representation Theory  Chapter 2.Linear Algebraic Groups over Arbitrary Ground Fields    1.Recollections from Algebraic Geometry     1.1.F-Structures on Affine Varieties     1.2.F-Structures on Arbitrary Varieties     1.3.Forms     1.4.Restriction of the Ground Field    2.F-Groups, Basic Properties     2.1.Generalities About F-Groups     2.2.Quotients     2.3.Forms     2.4.Restriction of the Ground Field    3.Tori     3.1.F-Tori     3.2.F-Tori in F-Groups     3.3.Split Tori in F-Groups    4.Solvable Groups     4.1.Solvable Groups     4.2.Sections     4.3.Elementary Unipotent Groups     4.4.Properties of Split Solvable Groups     4.5.Basic Results About Solvable F-Groups    5.Reductive Groups     5.1.Split Reductive Groups     5.2.Parabolic Subgroups     5.3.The Small Root System     5.4.The Groups G(F)     5.5.The Spherical Tits Building of a Reductive F-Group    6.Classification of Reductive F-Groups     6.1.Isomorphism Theorem     6.2.Existence     6.3.Representation Theory of F-Groups  Chapter 3.Special Fields    1.Lie Algebras of Algebraic Groups in Characteristic Zero     1.1.Algebraic Subalgebras    2.Algebraic Groups and Lie Groups     2.1.Locally Compact Fields     2.2.Real Lie Groups    3.Linear Algebraic Groups over Finite Fields     3.1.Lang's Theorem and its Consequences     3.2.Finite Groups of Lie Type     3.3.Representations of Finite Groups of Lie Type    4.Linear Algebraic Groups over Fields with a Valuation     4.1.The Apartment and Affine Dynkin Diagram     4.2.The Affine Building     4.3.Tits System, Decompositions     4.4.Local Fields    5.Global Fields     5.1.Adele Groups     5.2.Reduction Theory     5.3.Finiteness Results     5.4.Galois Cohomology  ReferencesII.Invariant TheoryAutbor IndexSubject Index

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