Modular representations of algebraic groups parshall, b. I took the construction of homogenous spaces ghand the proof that morphisms of linear algebraic groups have closed image from appendix a. Buy linear algebraic groups graduate texts in mathematics on free shipping on qualified orders. We extend borels theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. The other two standard references are the books with the same name by springer and borel. Lie groups, lie algebras, and representations, brian c. The text of the first edition has been corrected and revised. Quaternionic grassmannians and borel classes in algebraic. We then show that a complex connected reductive linear algebraic group is exponential if and. All algebraic geometry is over a fixed algebraically closed field. To do so, we rst study actions of algebraic groups. The equation c 0 defines the standard borel subgroup b of.
Mahdi asgari at the oklahoma state university in fall 2015. Classes of unipotent elements in simple algebraic groups. Humphreys before this, he held the posts of assistant professor of mathematics at the university of oregon and associate professor of mathematics at new york university. Given a 2, there exists an automorphism r a of v preserving such that r aa. Compact subgroups of linear algebraic groups sciencedirect. Rationality properties of linear algebraic groups, proc symp. Pdf ppt, program demos via download mathematics books collection or any other file from books category.
If youre interested in the theory of linear algebraic groups, linear algebraic groups by humphreys is a great book. A nite subset of v is a root system if it satis es. All of the algebraic geometry you need to know is built from scratch in any of those books. As a result, the student should be wellprepared to encounter groups, rings and elds in future courses in algebra, or other areas of discrete mathematics. The theory of borel equivalence relations studies the settheoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems. Numerous examples from algebraic groups, qschur algebras, and quantum groups are considered in detail. Inseparable extensions of algebras over the steenrod algebra with applications to modular invariant theory of finite groups ii neusel, mara d. They appear as certain projectivized orbit closures for some linear actions of reductive algebraic groups. This revised, enlarged edition of linear algebraic groups 1969 starts by presenting foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces.
For a linear algebraic group g over an algebraically closed field k, a borel subgroup of g means a maximal smooth connected solvable subgroup. We extend classical density theorems of borel and danishalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate. The main idea is a new approach to the structure of rank 1 groups. Word maps on perfect algebraic groups international journal. In section 4, we study borel subgroups of arbitrary linear algebraic groups. In another direction, we generalize borels theorem to some words with constants. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. Algebraic groups are groups which are algebraic varieties and such that the multiplication and the inverse maps are morphisms of algebraic varieties.
Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces. Springer have made a bunch of books available for free. It then turns to solvable groups, general properties of linear algebraic groups, and chevallys structure theory of reductive groups over algebraically closed groundfields. Borel subalgebra, quantum groups, linear algebra, homological functors. We also explain how to make a quotient ghinto a linear algebraic group when his a closed normal subgroup of g. Other readers will always be interested in your opinion of the books youve read.
Moreover, our big theorem about borel subgroups will only apply when g is nice, in a particular way. 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. While this may contradict the experience of many experienced mathematicians, the approach here is consciously algebraic. Linear algebraic groups university of toronto math department. Humphreys springerverlag new york wikipedia citation please see wikipedias template documentation for further citation fields that may be required. This book is a revised and enlarged edition of linear algebraic groups. 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. The sequel course \ linear algebraic groups ii treats the borel tits structure theory of reductive groups. Linear algebraic groups and countable borel equivalence.
Download pdf linear algebraic groups and finite groups of. 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. 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. Get unlimited access to the best stories on medium and support writers while youre at. Reductive group schemes, autour des schemas en groupes pdf, 1, paris. From 2 on, the word linear will usually be dropped, since more general algebraic groups will not be considered here. Hall lie groups, lie algebras and their representations linear algebra, werner greub. The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field.
The borel subgroups are important for the structure theory of linear algebraic groups. Here the group b is a borel subgroup and n is the normalizer of a maximal torus contained in b. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient. Despite being rooted in algebraic geometry, the subject has a fair mix of non algebraic geometric arguments. Linear algebraic groups oklahoma state universitystillwater. Some properties of adele groups attached to algebraic. Linear algebraic groups graduate texts in mathematics. Rationality properties of linear algebraic groups, ii. Elementary reference for algebraic groups mathoverflow. In the theory of algebraic groups, a borel subgroup of an algebraic group g is a maximal zariski closed and connected solvable algebraic subgroup.
Linear algebraic groups, graduate texts in mathematics, 126, springerverlag. In organizing this study, the following concept of reducibility is fundamental. Let e,f be equivalence relations on standard borel spaces x, y, resp. Algebraic groups play much the same role for algebraists as lie groups play for analysts. In mathematics, a linear algebraic group is a subgroup of the group of invertible n. 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. Borel subalgebras redux with examples from algebraic and.
The rational equivariant borelmoore homology groups as well as the rational. Springer have made a bunch of books available for free, here. 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. 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. Humphreys before this, he held the posts of assistant professor of mathematics at the university of oregon and associate professor of mathematics at.
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. Jul 04, 2007 some properties of adele groups attached to algebraic groups. Along the way, we will prove that linear algebraic groups can be embedded into some gl nsee corollary 4. If you notice any mistakes or have any comments, please let me know.
Springer have made a bunch of books available for free, here are the direct links. Lectures on the theory of algebraic numbers, erich hecke. This book is a revised and enlarged edition of linear algebraic groups, published by w. G always denotes a connected linear algebraic group with lie algebra g, t a maximal torus, and b a borel subgroup containing it. 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. Word maps on perfect algebraic groups international.
We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. Borel subalgebras come about by abstracting the theory of schubert varieties. Springer have made a bunch of books available for free, here are the direct links springerfreemathsbooks. For a linear algebraic group g over an algebraically closed field k, a borel subgroup. Algebraic groups play an important role in many areas of mathematics especially in geometry and representation theory.