This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. Elementary reference for algebraic groups mathoverflow. The first chapter lie algebras is mostly complete, the second algebraic groups treats only semisimple groups in detail, the third lie groups has yet to be written, and the appendix a survey of arithmetic subgroups is complete. Chevalieys seminar 14, seminar sophus lie 15 and monographs by c. Given a 2, there exists an automorphism r a of v preserving such that r aa. Roughly, the theory of linear algebraic groups generalizes that of linear lie groups over the real. This book is a revised and enlarged edition of linear algebraic groups, published by w.
Is there a book on linear algebraic groups using the scheme language i. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. The following books were the main sources for the class. Request pdf linear algebraic groups and countable borel equivalence relations this paper is a contribution to the study of borel equivalence relations on standard borel spaces i. Reductive groups these notes are a guide to algebraic groups, especially reductive groups, over a field. Borel subgroup, which is a maximal closed connected solvable subgroup. The remainder of the book is devoted to rationality questions over nonalgebraically. Our guiding idea was to present in the most economic way the theory of semisimple lie groups on the basis of the theory of algebraic groups. Second enlarged edition, graduate texts in mathematics 126, springerverlag, 1991 j.
Along the way, we will prove that linear algebraic groups can be embedded into some gl nsee corollary 4. The sequel course \linear algebraic groups ii treats the boreltits structure theory of reductive groups. Essays in the history of lie groups and algebraic groups. And similarly for g 2 suppose b is a borel subgroup containing t. Merger 142 references for chapter vi 144 chapter vii. Some of the material has not been dealt with before in other texts, such as rosenlichts results about solvable groups in chapter 14, the theorem of borel and tits on the conjugacy over the ground field of maximal split tori in an arbitrary linear algebraic group in chapter 15, and the tits classification of simple groups over a ground field in. Modulus of a rational map into a commutative algebraic group kato, kazuya and russell, henrik, kyoto journal of mathematics, 2010. The work of chevalley in lie groups and algebraic groups 147 1. For example, in the general linear group gl n n x n invertible matrices, the subgroup of invertible upper triangular matrices is a borel subgroup for groups realized over algebraically closed fields, there is a single conjugacy. To do so, we rst study actions of algebraic groups. These are notes for a class on algebraic groups taught by rajesh kulkarni at michigan state university during summer 2019. Within gare maximal, connected, solvable subgroups, called the borel subgroups.
Humphreys is presently professor of mathematics at the university of massachusetts at amherst. We also explain how to make a quotient ghinto a linear algebraic group when his a closed normal subgroup of g. Linear algebraic groups entirely avoids the use of scheme theory. The concept of a linear algebraic group may be introduced in two equivalent ways. May, 1975 linear algebraic groups entirely avoids the use of scheme theory. Second enlarged edition, graduate texts in mathematics 126, springerverlag, 1991. Book on linear algebraic groups in scheme language mathoverflow. Some of the material has not been dealt with before in other texts, such as rosenlichts results about solvable groups in chapter 14, the theorem of borel and tits on the conjugacy over the ground field of maximal split tori in an arbitrary linear algebraic group in chapter 15, and the tits classification of simple groups over a ground field in chapter 17. A fundamental fact is that the borels are conjugate in g, and much of the structure of gis grounded in those of the b. The borel subgroups containing a given maximal torus.
Humphreys, linear algebraic groups, graduate texts in. There are a number of analogous results between algebraic groups and coxeter groups for instance, the number of elements of the symmetric group is. When borel, chevalley, and others introduced algebraic geometry into the theory of algebraic groups, the foundations they used were those of the period e. Linear algebraic groups and finite groups of lie type, g. This revised, enlarged edition of linear algebraic groups 1969 starts by presenting foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces. Parabolic subgroups and flag varieties so far, weve been studying klein geometry, in particular projective geometry with symmetry group gln. This is a rough preliminary version of the book published by cup in 2017, the final version is substantially rewritten, and the numbering has changed. A nite subset of v is a root system if it satis es. Much of the theory of algebraic groups was developed. Linear algebraic groups modern birkhauser classics. Buy linear algebraic groups graduate texts in mathematics on free shipping on qualified orders linear algebraic groups graduate texts in mathematics. Reading that book, many people entered the research field of linear algebraic groups. The book should discuss topics like borel subgroups etc. In g glnk, a borel subgroup b is given by the upper triangular matrices.
Moreover, our big theorem about borel subgroups will only apply when g is nice, in a particular way. G always denotes a connected linear algebraic group with lie algebra g, t a maximal torus, and b a borel subgroup containing it. Accordingly, this book presents foundational material on algebraic groups, lie. Applications of borels theorem on borel subgroups 1. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient. Motivation in class we proved the important theorem of borel that if gis a connected linear algebraic group over an algebraically closed eld then gk is covered by the groups bk as bvaries through the borel subgroups of g. The other two standard references are the books with the same name by springer and borel. Thus, it is important to study solvable algebraic groups. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces. In the theory of algebraic groups, a borel subgroup of an algebraic group g is a maximal zariski closed and connected solvable algebraic subgroup. Chapter 8, geometry of the variety of borel subgroups. Algebraic groups play an important role in many areas of mathematics especially in geometry and representation theory.
Linear algebraic groups and countable borel equivalence. For example, in the general linear group gl n n x n invertible matrices, the subgroup of invertible upper triangular matrices is a borel subgroup. Algebraic groups are groups which are algebraic varieties and such that the multiplication and the inverse maps are morphisms of algebraic varieties. In g gl nk, a borel subgroup bis given by the upper triangular matrices. We assume the structure theory for connected solvable groups, and the completeness of the flag variety gb and some of its consequences. More specifically, it is an exposition of the theory of group schemes of. Conjugacy classes in algebraic groups, steinberg 6 linear algebraic groups, t. We will only consider the category of affine algebraic groups, a. From 2 on, the word linear will usually be dropped, since more general algebraic groups will not be considered here. There are several good books on algebraic groups like bor69. After establishing these basic topics, the text then turns to solvable groups, general properties of linear algebraic groups and chevallys structure theory of reductive groups over algebraically closed groundfields. In section 4, we study borel subgroups of arbitrary linear algebraic groups. An important ingredient in studying linear algebraic groups is the notion of a.
The sequel course \ linear algebraic groups ii treats the borel tits structure theory of reductive groups. The text of the first edition has been corrected and revised. Linear algebraic groups without the normalizer theorem 5 4 ngt containsanelementactingont byinversion. The elements gs and gu are called the semisimple and unipotent part of g, respectively. Groups of isometrics on operator algebra ii pedersen, steen, rocky mountain journal of mathematics, 1988. Chapter 10, representations of semisimple algebraic. Likewise, for a linear algebraic group g over c, gc is a complex lie group. All algebraic geometry is over a fixed algebraically closed field. An accessible text introducing algebraic groups at advanced undergraduate and early graduate level, this book covers the conjugacy of borel subgroups and maximal tori, the theory of algebraic groups with a bnpair, frobenius maps on affine varieties and algebraic groups, zeta functions and lefschetz numbers for varieties over finite fields. Springer 5 linear algebraic groups, humphreys 3 linear algebraic groups, borel 2. Humphreys mentions in the preface that part of the motivation to write the textbook in the first place was the lack of an elementary treatment of the subject. Linear algebraic groups graduate texts in mathematics. If youre interested in the theory of linear algebraic groups, linear algebraic groups by humphreys is a great book. This book is a revised and enlarged edition of linear algebraic groups.
I took the construction of homogenous spaces ghand the proof that morphisms of linear algebraic groups have closed image from appendix a. Linear algebraic groups i stanford, winter 2010 notes typed by sam lichtenstein, lectures and editing by brian conrad february 8, 2020 please send any errata typos, math errors, etc. For a linear algebraic group g over the real numbers r, the group of real points gr is a lie group, essentially because real polynomials, which describe the multiplication on g, are smooth functions. We will also use some basic facts on lie algebras and probably use root systems but i will probably recall at least the basic definitions on that subject. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups.
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