Linear algebraic groups exceptionally wellwritten and ideally suited either for independent reading or as a graduate level text for an introduction to everything about linear algebraic groups. Springer made a bunch of books available for free, these. Springer made a bunch of books available for free, these were. The book should discuss topics like borel subgroups etc. The galois groups in this galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. Questions about the book linear algebraic groups by springer. Linear representations of finite groups, jeanpierre serre. Springer 5 linear algebraic groups, humphreys 3 linear algebraic groups, borel 2. Armand borel, linear algebraic groups, springer gtm, 2 editions tonny a. Clark it is an initially surprising fact how much of the geometry and arithmetic of shimura varieties e. Springer have made a bunch of books available for free. 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.
Linear algebraic groups graduate texts in mathematics. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient. Humphreys, linear algebraic groups, graduate texts in mathematics, springer. A linear algebraic group over an algebraically closed field k is a subgroup of a group gl n k of invertible n. Second edition, progress in mathematics 9, birkhauser, 1998. Springer, rationality properties of linear algebraic groups, proc. Donkin, s good filtrations of rational modules for reductive groups. This book is a revised and enlarged edition of linear algebraic groups, published by w. Lie groups, lie algebras, and representations, brian c.
First typo that i came across is in page 6, regular functions and ringed spa. There are several good books on algebraic groups like bor69. His main research interests include group theory and lie algebras, and this graduate level text is an exceptionally wellwritten introduction to everything about linear. Errata for linear algebraic groups by springer mathoverflow. Linear differential equations form the central topic of this volume, galois theory being the unifying theme. In linear algebraic groups springer aims at a selfcontained treatment of the subject in the title and he certainly succeeds. 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. Springerverlag published his first book, introduction to lie algebras and representation theory graduate texts in mathematics vol. There are two important classes of algebraic groups whose intersection is trivial the identity group. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra.
Any algebraic group contains a unique normal linear algebraic subgroup h h such that their quotient g h gh is an abelian variety. Introduction to linear algebraic groups michel brion. We extend and apply the galois theory of linear differential equations equipped with the action of an endomorphism. Despite being rooted in algebraic geometry, the subject has. Chapter 10, representations of semisimple algebraic groups. Homework will be assigned from time to time in lecture. Likewise, for a linear algebraic group g over c, gc is a complex lie group. The first three books borel, humphreys, springer are the classical textbooks on. Quadratic forms, linear algebraic groups, and cohomology. Linear algebraic groups s if q c, every affine algebraic group g can be viewed as a complex lie group. And all of it is done in a very clear style, making for a smooth and readable presentation. Reading that book, many people entered the research field of linear algebraic groups. From 2 on, the word linear will usually be dropped, since more general algebraic groups will not be considered here. The text of the first edition has been corrected and revised.
Linear algebraic groups modern birkhauser classics. Modular representations of algebraic groups parshall, b. Roughly speaking a linear algebraic group is a subgroup of a group of matrices which is defined by algebraic conditions. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. Linear algebraic groups, graduate texts in mathematics 126, springerverlag 1991. His main research interests include group theory and lie algebras. He did graduate work in philosophy and mathematics at cornell university and later received hi ph. Linear algebraic groups, graduate texts in mathematics, 21. Accordingly, this book presents foundational material on algebraic groups, lie. Springer, linear algebraic groups, progress in mathematics, 9 2nd ed. This is a quick introduction to the main structural results for affine algebraic groups over algebraically closed fields with full proofs but assuming only a very modest background. Linear operators in hilbert spaces, joachim weidmann. In 1972, springerverlag published his first book, introduction to lie algebras and representation theory graduate texts in mathematics vol. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field.
Is there a book on linear algebraic groups using the scheme language i. An algebraic introduction emphasis on the theory of graphs. He has previously held posts at the university of oregon and new york university. Introduction to linear algebraic groups universite grenoble alpes. Vinberg, lie groups and algebraic groups, springerverlag, 1990 and the notes available on the authors web page t. Humphreys is presently professor of mathematics at the university of massachusetts at amherst. If v has finite dimension n, then glv and gln, f are isomorphic. 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.
The necessary techniques from algebraic geometry are developed from scratch along the way. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Linear algebraic groups entirely avoids the use of scheme theory. Keywords algebraic algebraische gruppe group theory lie algebra representation theory algebra. Algebraic groups play much the same role for algebraists as lie groups play for analysts. His main research interests include group theory and lie algebras, and this graduate level text is an exceptionally wellwritten introduction to everything about linear algebraic groups. A subgroup of g of gln, q is called an algebraic matrix group if g is a. Springer have made a bunch of books available for free, here. The following books were the main sources for the class. Does any one prepared a list of errata for linear algebraic groups by springer. I am reading the book linear algebraic groups by springer.
Lectures on the theory of algebraic numbers, erich hecke. This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. An example is the orthogonal group, defined by the relation m t m 1 where m t is the transpose of m. If v is a vector space over the field f, the general linear group of v, written glv or autv, is the group of all automorphisms of v, i. Much of the theory of algebraic groups was developed.
Despite being rooted in algebraic geometry, the subject has a fair mix of non algebraic geometric arguments. Springer published by springer berlin heidelberg isbn. Graduate seminar linear algebraic groups fall 2016 home. Book on linear algebraic groups in scheme language mathoverflow. 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. 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. Linear algebraic groups department of mathematics university of. May, 1975 linear algebraic groups entirely avoids the use of scheme theory.
Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces. I have a question on page 53, on line 3, it is said that. These are notes for a class on algebraic groups taught by rajesh kulkarni at michigan state university during summer 2019. In the old days when books were typically typeset by hand and publishers made fixed press runs which often created tax problems for unsold warehoused books, it was. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
A good reference for lie algebras is bou71, for root systems, see ser66 while we refer to har77 for algebraic geometry. Jones introduction to laplace transforms and fourier series p. Many lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. My o ce hours are monday 34 and tuesday 12, or by appointment. Tonny albert springer february 1926 7 december 2011 was a mathematician at utrecht university who worked on linear algebraic groups, hecke algebras, complex reflection groups, and who introduced springer representations and the springer resolution. More specifically, it is an exposition of the theory of group schemes of. Tonny albert springer february 1926 7 december 2011 was a mathematician at utrecht university who worked on linear algebraic groups, hecke algebras, complex reflection groups, and who introduced springer representations and the springer resolution springer began his undergraduate studies in 1945 at leiden university and remained there for his graduate work in. 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. There are a number of analogous results between algebraic groups and coxeter groups for instance, the number of elements of the symmetric group is. 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. Linear algebraic groups and finite groups of lie type, g.
Humphreys is a distinguished professor of mathematics at the university of massachusetts at amherst. Hall lie groups, lie algebras and their representations linear algebra, werner greub. Michel brion universit e grenoble alpes master ii degree in pure math september 4th, 2017 8 8. In 1972, springerverlag published his first book, introduction to lie algebras.