A semigroup approach to linear algebraic groups ii. Uc berkeley summer undergraduate research program 2015 july 1. The belief that in algebraic geometry an affine space means a set with n linear coordinate functions defined on it simply does not agree with sources, nor indeed with any kind of reasonable common sense. Buy linear algebraic groups graduate texts in mathematics on free shipping on qualified orders linear algebraic groups graduate texts in mathematics. Linear algebra is the study of vectors and linear functions. It turns out that just about any set you can describe on the real line is a borel set, for example, the unit line segment 0. Reading that book, many people entered the research field of linear algebraic groups.
Uc berkeley summer undergraduate research program 2015 july 1 lecture we are going to introduce some of the basic structure of the general linear group glv, where v is a nite dimensional cvector space. I took the construction of homogenous spaces ghand the proof that morphisms of linear algebraic groups have closed image from appendix a. Putcha department of mathematics, north carolina state university, raleigh, nc 276958205, usa communicated by j. If iis prime, then v i is an a ne algebraic variety. An example is the orthogonal group, defined by the relation m t m 1 where m t is the transpose of m.
Linear algebraic groups graduate texts in mathematics. In section 4, we study borel subgroups of arbitrary linear algebraic groups. 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. Linear algebraic groups and finite groups of lie type. Linear algebraic groups over arbitrary fields and its related structures, like azumaya algebras, algebras with involutions, brauer groups, quadratic and hermitean forms, witt rings, lie and jordan algebras, homogeneous varieties some related manuscripts are to be found on. 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. Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. The other two standard references are the books with the same name by springer and borel. Aug 20, 20 for the love of physics walter lewin may 16, 2011 duration. Knapp, basic algebra, digital second edition east setauket, ny. Linear algebraic groups oklahoma state universitystillwater.
I have a copy of linear algebraic groups by humphreys. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The aim of the old book was to present the theory of linear algebraic groups over an algebraically closed field. This is in some sense unfortunate, because the theory of alge. The sequel course \ linear algebraic groups ii treats the borel tits structure theory of reductive groups. Mahdi asgari at the oklahoma state university in fall 2015. To do so, we rst study actions of algebraic groups.
While this may contradict the experience of many experienced mathematicians, the approach here is consciously algebraic. Elementary reference for algebraic groups mathoverflow. A measure theory tutorial measure theory for dummies. Borel subalgebra, quantum groups, linear algebra, homological functors.
A nite subset of v is a root system if it satis es. Uc berkeley summer undergraduate research program 2015. Modular representations of algebraic groups parshall, b. Despite being rooted in algebraic geometry, the subject has a fair mix of non algebraic geometric arguments. If youre interested in the theory of linear algebraic groups, linear algebraic groups by humphreys is a great book. A good reference for lie algebras is bou71, for root systems, see ser66 while we refer to har77 for algebraic geometry.
Along the way, we will prove that linear algebraic groups can be embedded into some gl nsee corollary 4. 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. 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. Borel subalgebras come about by abstracting the theory of schubert varieties. Find all the books, read about the author, and more. Also, check out milnes algebraic geometry or borels linear algebraic groups, where they do things in a coordinateinvariant way. Also, the normalizer, the center, and the centralizer of a closed subgroup h of a linear algebraic group g are naturally viewed as closed subgroup schemes of g. We also explain how to make a quotient ghinto a linear algebraic group when his a closed normal subgroup of g.
Algebraic groups lecture notes uw courses web server. The text of the first edition has been corrected and revised. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient spaces. A linear algebraic group, g, is a arievty vkwith a group structure such that the group. Linear algebraic groups fall 2015 these are notes for the graduate course math 6690 linear algebraic groups taught by dr. Journal of pure and applied algebra 39 1986 153163 153 northholland a semigroup approach to linear algebraic groups ii.
In broad terms, vectors are things you can add and linear functions are functions of vectors that respect vector addition. This preprint server is intended to be a forum of the recent development of the theory of. From 2 on, the word linear will usually be dropped, since more general algebraic groups will not be considered here. The study of these groups and algebraic group homomorphisms among them which can be defined in an obvious way goes under the name linear algebraic groups and their representations. So i am wondering where in linear algebraic groups i can find the related result or proof. Lectures delivered at the university of notre dame, ed. There are several good books on algebraic groups like bor69. In mathematics, a linear algebraic group is a subgroup of the group of invertible n. Numerous examples from algebraic groups, qschur algebras, and quantum groups are considered in detail.
Many lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. Its aim is to treat the theory of linear algebraic groups over arbitrary fields. Borel subalgebras redux with examples from algebraic and. Accordingly, this book presents foundational material on algebraic groups, lie algebras, transformation spaces, and quotient. Clark it is an initially surprising fact how much of the geometry and arithmetic of shimura varieties e. And properties of the symplectic group spv, if theres time. Knapp, 2016, 2016 admissible monomials and generating sets for the polynomial algebra as a module over the steenrod algebra mothebe, m. Here, and in the following, topological notions such as closedness. Moreover, our big theorem about borel subgroups will only apply when g is nice, in a particular way. The goal of this text is to teach you to organize information about vector spaces in a way that makes problems involving linear functions of many variables easy. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. Knapp, advanced algebra, digital second edition east setauket, ny.
Do the borel subgroups of an algebraic group coincide with. For example, a linear algebraic group is solvable if it has a composition series of linear algebraic subgroups such that the quotient groups are commutative. Finite groups of lie type and their representations 5 1. David cherney, tom denton, rohit thomas and andrew waldron. A linear algebraic group is called linearly reductive if every. All of the algebraic geometry you need to know is built from scratch in any of those books.
The sequel course \linear algebraic groups ii treats the boreltits structure theory of reductive groups. Despite being rooted in algebraic geometry, the subject has a fair mix of nonalgebraic geometric arguments. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. To make ginto an algebraic group, we have to give a. 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. Keith linear algebra, 2nd edition kenneth hoffmann and ray kunze linear algebraic groups 2ed borel 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. Milgram notre dame, ind university of notre dame, 1942, 1942. But i cant find conjugacy classes in algebraic groups. Rhodes received august 1983 introduction linear algebraic monoids arise naturally in connection with the representations of a.
If you notice any mistakes or have any comments, please let me know. In the theory of algebraic groups, a borel subgroup of an algebraic group g is a maximal zariski closed and connected solvable algebraic subgroup. Given a 2, there exists an automorphism r a of v preserving such that r aa. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. Parabolic subgroups and flag varieties so far, weve been studying klein geometry, in particular projective geometry with symmetry group gln. Roughly, the theory of linear algebraic groups generalizes that of linear lie groups over the real. This purity condition is part of the formulation in prop. For the love of physics walter lewin may 16, 2011 duration. Also, check out milnes algebraic geometry or borel s linear algebraic groups, where they do things in a coordinateinvariant way. It is a theorem that all affine algebraic groups are in fact linear algebraic groups. He has previously held posts at the university of oregon and new york university. This book is a revised and enlarged edition of linear algebraic groups, published by w. 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 class of borel. This course aims to introduce students with a background in basic algebra, commutative algebra, and topologybut not necessarily one in algebraic geometryto the subject of linear algebraic groups.
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