Borel subgroups are one of the two key ingredients in understanding the structure of simple more generally, reductive algebraic groups, in jacques tits theory of groups with a b,n pair. The sequel course \ linear algebraic groups ii treats the borel tits structure theory of reductive groups. Rationality properties of linear algebraic groups and galois. Is there a book on linear algebraic groups using the scheme language i. Kechris this paper is a contribution to the study of borel equivalence relations on standard borel spaces i. On the failure of the gorenstein property for hecke algebras of prime weight kilford, l. Our most explicit results concern rootreductive indgroups g, the. Chapter 8, geometry of the variety of borel subgroups. For a linear algebraic group g over an algebraically closed field k, a borel subgroup of g means a maximal smooth connected solvable subgroup.
From 2 on, the word linear will usually be dropped, since more general. Despite being rooted in algebraic geometry, the subject has a fair mix of non algebraic geometric arguments. The concept of a linear algebraic group may be introduced in two equivalent ways. I took the construction of homogenous spaces ghand the proof that morphisms of linear algebraic groups have closed image from appendix a. Gis a linear algebraic group over k, kalgebraically closed. Borel subalgebra, quantum groups, linear algebra, homological functors. We will only consider the category of affine algebraic groups, a. The work of chevalley in lie groups and algebraic groups 147 1. Reductive groups these notes are a guide to algebraic groups, especially reductive groups, over a field.
Spiegel, eugene and trojan, allan, pacific journal of mathematics, 1976. Linear algebraic groups form a broad generalization of the classical groups. 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. Introduction to linear algebraic groups michel brion universit e grenoble alpes september 4th, 2017 michel brion universit e grenoble alpes master ii degree in pure math september 4th, 2017 1 8. Given a 2, there exists an automorphism r a of v preserving such that r aa. Linear algebraic groups without the normalizer theorem arxiv. Find all the books, read about the author, and more. The sequel course \linear algebraic groups ii treats the boreltits structure theory of reductive groups.
Linear algebraic groups mat 1110, winter 2017 homework 4, due march 29, 2017 florian herzig problem 1. Applications of borels theorem on borel subgroups 1. 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. The book should discuss topics like borel subgroups etc. Suppose that g is a connected linear algebraic group. The other two standard references are the books with the same name by springer and borel. Borel subgroup, which is a maximal closed connected solvable subgroup. The variety of tori, unirationality of reductive groups. A bottborelweil theory for direct limits of algebraic groups. Here the group b is a borel subgroup and n is the normalizer of a maximal torus contained in b. Linear algebraic groups and finite groups of lie type, g. The borel subgroups are important for the structure theory of linear algebraic groups. Inseparable extensions of algebras over the steenrod algebra with applications to modular invariant theory of finite groups ii neusel, mara d.
Linear algebraic groups fall 2015 these are notes for the graduate course math 6690 linear algebraic groups taught by dr. Linear algebraic groups graduate texts in mathematics 2nd edition. Borel subalgebras redux with examples from algebraic and. Merger 142 references for chapter vi 144 chapter vii. Secondly, we are interested in the galois cohomology of linear algebraic groups and also. Other readers will always be interested in your opinion of the books youve read. Chapter 10, representations of semisimple algebraic. A nite subset of v is a root system if it satis es. Springer have made a bunch of books available for free, here are the direct links springerfreemathsbooks. More specifically, it is an exposition of the theory of group schemes of. An important ingredient in studying linear algebraic groups is the notion of a.
Linear algebraic groups florian herzig university of toronto math. All of the algebraic geometry you need to know is built from scratch in any of those books. This book is a revised and enlarged edition of linear algebraic groups. Algebraic groups play much the same role for algebraists as lie groups play for analysts. The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. Many lie groups can be viewed as linear algebraic groups over the field of real or complex numbers.
Despite being rooted in algebraic geometry, the subject has a fair mix of nonalgebraic geometric arguments. 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. Along the way, we will prove that linear algebraic groups can be embedded into some gl nsee corollary 4. Algebraic groups are groups which are algebraic varieties and such that the multiplication and the inverse maps are morphisms of algebraic varieties. Bis borel if and only if it a minimal parabolic subgroup. 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. Closed orbits of borel subgroups to cite this article. Let us recall a few facts about complete varieties. Groups of isometrics on operator algebra ii pedersen, steen, rocky mountain journal of mathematics, 1988. If you notice any mistakes or have any comments, please let me know.
We develop a bottborelweil theory for direct limits of algebraic groups. Linear algebraic groups and countable borel equivalence relations article in journal of the american mathematical society 4 july 2000 with 10 reads how we measure reads. This text is relatively selfcontained with fairly standard treatment of the subject of linear algebraic groups as varieties over an algebraic closed field not necessarily characteristic 0. To do so, we rst study actions of algebraic groups. Essays in the history of lie groups and algebraic groups. Second enlarged edition, graduate texts in mathematics 126, springerverlag, 1991.
Modular representations of algebraic groups parshall, b. In the theory of algebraic groups, a borel subgroup of an algebraic group g is a maximal zariski closed and connected solvable algebraic subgroup. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 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. Moreover, our big theorem about borel subgroups will only apply when g is nice, in a particular way.
Rationality properties of linear algebraic groups and galois cohomology mcm, march 2007 philippe gille the goal of this series if lecture is. 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. Linear algebraic groups department of mathematics university of. They show up in various domains of mathematics such as algebra in particular, di erential galois theory. Algebraic groups play an important role in many areas of mathematics especially in geometry and representation theory. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces. Linear algebraic groups and countable borel equivalence. Mahdi asgari at the oklahoma state university in fall 2015. Linear algebraic groups oklahoma state universitystillwater. Borel subalgebras come about by abstracting the theory of schubert varieties.
Linear algebraic groups graduate texts in mathematics. 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. Numerous examples from algebraic groups, qschur algebras, and quantum groups are considered in detail. Modulus of a rational map into a commutative algebraic group kato, kazuya and russell, henrik, kyoto journal of mathematics, 2010. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Lie groups and linear algebraic groups by professor a.
Some of our results apply to locally reductive indgroups g in general, i. From 2 on, the word linear will usually be dropped, since more general algebraic groups will not be considered here. Fundamental hermite constants of linear algebraic groups watanabe, takao, journal of the mathematical society of japan, 2003. When borel, chevalley, and others introduced algebraic geometry into the theory of algebraic groups, the foundations they used were those of the period e. 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. An example is the orthogonal group, defined by the relation m t m 1 where m t is the transpose of m. The borel subgroups containing a given maximal torus. The main idea is a new approach to the structure of rank 1 groups. G always denotes a connected linear algebraic group with lie algebra g, t a maximal torus, and b a borel subgroup. Springer have made a bunch of books available for free.
We also explain how to make a quotient ghinto a linear algebraic group when his a closed normal subgroup of g. 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. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient. In section 4, we study borel subgroups of arbitrary linear algebraic groups.