Algebraic groups are groups which are algebraic varieties and such that the multiplication and the inverse maps are morphisms of algebraic varieties. His main research interests include group theory and lie algebras. These notes are a comprehensive modern introduction to the theory of algebraic groups assuming only the knowledge of algebraic geometry usually acquired in a. 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. 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. That is, gis a group and a variety, and the maps g.
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. I merely distill some of the material from humphreys and springer. The coordinate ring kg of a linear algebraic group is a commutative. Linear algebraic groups s if q c, every affine algebraic group g can be viewed as a complex lie group. Pseudoreductive groups arise naturally in the study of general smooth linear algebraic groups over nonperfect fields and have many important applications. Given a 2, there exists an automorphism r a of v preserving such that r aa. He has previously held posts at the university of oregon and new york university. Much of the theory of algebraic groups was developed. Humphreys, linear algebraic groups, graduate texts in mathematics, springer. Linear algebraic groups, graduate texts in mathematics, 21. There are a number of analogous results between algebraic groups and coxeter groups for instance, the number of elements of the symmetric group is. Springer graduate text gtm 173 by reinhard diestel covers linear algebra on graphs 2012, p. For such a group glet kg denote the algebra of regular functions on g.
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. Linear representations of finite groups, jeanpierre serre. For instance, a linear algebraic group is irreducible as a variety if and only if it is connected see proposition 1. In this lecture we shall present the basic theory of algebraic groups over any algebraically closed. Linear algebraic groups graduate texts in mathematics. This book is a revised and enlarged edition of linear algebraic groups, published by w. This monograph provides a comprehensive treatment of the theory of pseudoreductive groups and gives their classification in a usable form. 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. 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. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or lie groups, and it is not just the groups themselves that are of interest, but also their linear representations. Linear algebraic groups, graduate texts in mathematics 126, springerverlag 1991. Springer made a bunch of books available for free, these were. More specifically, it is an exposition of the theory of group schemes of.
Thus linear algebraic groups can be viewed as certain groups of matrices. All of the algebraic geometry you need to know is built from scratch in any of those books. Linear operators in hilbert spaces, joachim weidmann. For the background on algebraic groups i refer to the books linear algebraic groups by j. Descargar linear algebraic groups en pdf libros geniales.
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. A nite subset of v is a root system if it satis es. Linear algebraic groups graduate texts in mathematics pdf. Groups and algebras of binary relations givant, steven and andreka, hajnal, bulletin of symbolic logic, 2002. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces. Humphreys is presently professor of mathematics at the university of massachusetts at amherst. Introduction to lie algebras and representation theory. The sequel course \ linear algebraic groups ii treats the boreltits structure theory of reductive groups. If youre interested in the theory of linear algebraic groups, linear algebraic groups by humphreys is a great book. Humphreys, linear algebraic groups, graduate texts in mathematics, v olume 21, springer, 1975. 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. A linear algebraic group over an algebraically closed field k is a subgroup of a group gl n k of invertible n.
Here i merely distill some of the material from humphreys and springer. Algebraic groups and class fields jeanpierre serre auth. Elliptic curves two important classes of algebraic groups arise, that for the most part are studied separately. Humphreys is presently 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. Roughly, the theory of linear algebraic groups generalizes that of linear lie groups over the real. 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. As far as algebraic groups go, i have had a look at the books by humphreys, borel and tom springer as well as the notes of james milne. Linear algebraic groups department of mathematics university of. Springer made a bunch of books available for free, these. Borel subgroups and maximal tori of linear algebraic groups.
An example is the orthogonal group, defined by the relation m t m 1 where m t is the transpose of m. Pdf jordan property for nonlinear algebraic groups and. Representations of algebraic groups and their lie algebras. 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. If iis prime, then v i is an a ne algebraic variety. Humphreys is a distinguished professor of mathematics at the university of massachusetts at amherst. The text of the first edition has been corrected and revised. Several important classes of groups are algebraic groups, including. Humphreys before this, he held the posts of assistant professor of mathematics at the university of oregon and associate professor of mathematics at. Elementary reference for algebraic groups mathoverflow. The first three books borel, humphreys, springer are the classical textbooks on. A linear algebraic group, g, is a arievty vkwith a group structure such that the group. The necessary techniques from algebraic geometry are.
In mathematics, many of the groups that appear naturally have a nice description as matrix. Humphreys is presently professor of mathematics at the university of. Graduate seminar linear algebraic groups fall 2016. Lecture 7 5 we then choose a 2dimensional subspace v 2 with v 1 v 2, which is equivalent to choosing a 1dimensional subspace or line in knv 1. By dan saracino i havent seen any other book explaining the basic concepts of abstract algebra this beautifully.
In these lecture we shall concentrate on affine algebraic. Download pdf linear algebraic groups and finite groups. Gln, c, the general linear group of invertible matrices over cjet group. If h is a linear algebraic subgroup of g, then gh is also, a manifold, and dimgh dimg dimh. We show that reductive linear algebraic groups over some. His main research interests include group theory and lie algebras, and this graduate level text is an exceptionally wellwritten introduction to. Algebraic groups, lie groups, and their arithmetic subgroups this work has been replaced by the above three, and will not be revised or corrected. Ws1415 linear algebraic groups arbeitsgruppe algebra. Algebraic groups play an important role in many areas of mathematics especially in geometry and representation theory. The sequel course \linear algebraic groups ii treats the boreltits structure theory of reductive groups.
By convention all our algebraic groups will be linear algebraic groups over k. The other two standard references are the books with the same name by springer and borel. Algebraic groups play much the same role for algebraists as lie groups play for analysts. Pseudoreductive groups by brian conrad cambridge core. The arguments humphreys gives linear algebraic groups entirely avoids the use of scheme theory.
Geometric functional analysis and its applications. Chapter 10, representations of semisimple algebraic groups. 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. Algebraic groups and class fields jeanpierre serre.
Moreover, any linear algebraic group can be embedded as a closed subgroup of gl n for some n see corollary 4. Assuming only a good understanding of linear and abstract algebra, humphreys starts with the definition of a lie algebra and, about one hundred pages later, has gone through their basic properties and established the classification theorem for semisimple lie algebras over an algebraically closed field of characteristic 0. The elements gs and gu are called the semisimple and unipotent part of g, respectively. Despite being rooted in algebraic geometry, the subject has a fair mix of nonalgebraic geometric arguments.
Likewise, for a linear algebraic group g over c, gc is a complex lie group. A final project related to algebraic groups, suggestions for a final topic have been for example what is a reductive group. May, 1975 the arguments humphreys gives linear algebraic groups entirely avoids the use of scheme theory. James edward humphreys 1939, erie, pennsylvania is an american mathematician, who works on algebraic groups, lie groups, and lie algebras and applications of these mathematical structures.
Many lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. The necessary techniques from algebraic geometry are developed from scratch along the way. Graphs and matrices by bapat as pointed out by josse. He is known as the author of several mathematical texts, especially introduction to lie algebras and representation theory. 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. Pdf let g be a connected complex algebraic group and a an abelian connected algebraic group, together with an algebraic action of g on a via group. View mac lane categories for the working mathematician. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups.
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