It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of lie type are known. On the properly discontinuous subgroups of affine motions. Mostow algebraic groups and discontinuous subgroups, proc. Learn to identify subgroups of a group in general as well as in many powerful examples, like the group of permutations of a finite set and the group of symmetries of a regular polygon. Geometric group theory preliminary version under revision. The cayley table for h is the topleft quadrant of the cayley table for g. Functional analysis local field algebraic group these keywords were added by machine and not by the authors. This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. In general, subgroups of cyclic groups are also cyclic. Milnes algebraic groups and arithmetic groups defines an arithmetic subgroup of g q as a subgroup commensurable with the stabilizer of a lattice in a faithfull rational representation. 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. Arithmetic subgroups of algebraic groups by armand borel and harishchandra communicated by deane montgomery, july 22, 1961. Representations of finite groups pdf 75p download book.
For example, the additive group and the multiplicative group have the same 1dimensional lie algebra, which over a eld such as c fails to distinguish the two groups. Here are some fundamental facts about complete varieties. Classification of algebraic simple groups, algebraic groups and discontinuous subgroups, proc. Then, in section 6, we combine this characterization of. On combining these statements, we see that the ci form a complete set of orthogonal. Properly discontinuous actions of subgroups in amenable. These groups are algebraic groups, and we shall look only at representations g glv that are homomorphisms of algebraic groups. We combine his methods with the recent computations of torsion indexes. Algebraic groups and discontinuous subgroups group.
On irreducible subgroups of simple algebraic groups 1261 where the wi are pairwise orthogonal subspaces of w. Tori, unipotent and connected solvable groups 12 3. In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example they arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory. Arithmetic subgroups of algebraic groups by armand borel and harishchandra received october 18, 1961 table of contents. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. A note on quotients of real algebraic groups by arithmetic subgroups. Compact subgroups of linear algebraic groups by richard pink fakultat fu. Strongly dense free subgroups of semisimple algebraic groups 3 are contained in some. This volume collects articles that arose from this event, which took place at a time when algebraic geometry was undergoing a major change.
In algebraic geometry, an algebraic group or group variety is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety. The attendees ranged from people who have never studied algebraic groups to experts. A note on generators for arithmetic subgroups of algebraic groups. Browse other questions tagged abstract algebra group theory finite groups sylowtheory p groups or ask your own question. Spinors and essential dimension compositio mathematica. On irreducible subgroups of simple algebraic groups. We also explain how to make a quotient ghinto a linear algebraic group when his a closed normal subgroup of g.
The exposition of the topic is built on a synthesis of methods from algebraic geometry, number theory. On arithmetic subgroups of simple algebraic groups l. This process is experimental and the keywords may be. Algebraic groups play much the same role for algebraists as lie groups play for analysts. Currently this section contains no detailed description for the page, will update this page soon. We study the question of which torsion subgroups of commutative algebraic groups over. Groups are in some ways simpler, having just a single operation, but rings are more familiar since the integers make a good prototype to think about. A quasiprojective variety is complete if and only if it is projective. Algebraic groups and discontinuous subgroups author. Algebraic groups with few subgroups skip garibaldi and philippe gille abstract every semisimple linear algebraic group over a.
Algebraic groups and discontinuous subgroups free ebook download as pdf file. Barker abstract arithmetic subgroups of simple isotropic algebraic groups are described as subgroups full of root elements. A computational introduction to number theory and algebra victor shoups. Algebraic subgroups of lie groups heldermannverlag. Descargar algebraic groups and discontinuous subgroups. Jun 14, 2005 on algebraic groups and discontinuous groups. A subgroup h of a group gis said to be a normal subgroup of gif ghg 1 2hfor all h2hand g2g. It allows one to identify many groups with subgroups of gln, k, the group of the invertible matrices of dimension n over a field k.
A complex algebraic group g is in this note a subgroup of gln, c, the elements of which are all invertible matrices whose coefficients annihilate some set of polynomials pmxn, xnn in n2 indeterminates. Matsumoto, subgroups of finite index in certain arithmeticgroups, in sym posium on algebraic groups and discontinuous subgroups, amer. We do prove weaker results for all algebraically closed. Craven birmingham maximal subgroups 4th march, 2014 9 20. Discontinuous subgroups of pgl connecting repositories. On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces article in mathematical notes 895. Algebraic groups and discontinuous subgroups a borel, g. Maximal subgroups of exceptional groups of lie type and. Determining the subgroup structure of algebraic groups over an algebraically closed field k of arbitrary characteristic often requires an understanding of those instances when a group y and a. Then gis a group, and we can write out its multiplication table cayley table. Descargar algebraic groups and number theory en pdf. Algebraic groups and discontinuous subgroups, proc.
Later on, chapter 11 is devoted to a rapid development of linear algebraic groups, up to borel subgroups and maximal tori but without the classi. Let r be a discrete subgroup of g, k a maximal compact subgroup. We record some further nice properties of integral extensions. It was proved by platonov, 8, that the number of conjugacy classes of maximal solvable subgroups not necessarily connected in an algebraic group over an algebraically closed eld is nite. A group action on a finitedimensional vector space is called a representation of the group.
In this work, we introduce the notion of algebraic subgroups of complex lie groups, and prove that every faithfully representable complex analytic group g admits an algebraic subgroup tg which is the largest in the sense that it contains all algebraic. Along the way, we will prove that linear algebraic groups can be embedded into some gl nsee corollary 4. The authors of this book investigate the subgroups of certain types of simple algebraic groups and obtain a complete description of all those subgroups which are themselves simple. On the geometry of stable discontinuous subgroups acting. On arithmetic subgroups of simple algebraic groups. The goal of this book is to present several central topics in geometric group theory, primarily related to the large scale geometry of infinite groups and spaces on which such groups act, and to illustrate them with fundamental theorems such as gromovs theorem on groups of polynomial growth. We can combine theorem 19 i, corollary 5 ii and theorem 20 to show. In this survey, we will discuss how group representation theory helps us study this classical problem. In terms of category theory, an algebraic group is a group object in the category of algebraic varieties. Consequently the series began with two introductory talks on the structure of algebraic. Let a be a group that acts smoothly and properly discontinuously on a smooth contractible manifold x via a homomorphism p. We switch to the dual category of finitely generated abelian groups with a continuous action. Subgroup structure and representations of finite and.
The direct product is a way to combine two groups into a new, larger group. Algebra course fma190fma190f 2011, spring term 2011, sergei silvestrov lectures subgroups a subgroup h of a group g is a group contained in g so that if h, h02h, then the product hh0in h is the same as the product hh0in g. Another example is the union of the xaxis and the yaxis in the plane with the addition operation. The group of rigid motions of a regular ngon is called the nth dihedral group, denoted by d n. In section 4, we study borel subgroups of arbitrary linear algebraic groups. The results about maximal subgroups and morphisms of algebraic groups for an embedding hq into g depend on whether h has large, medium or small rank, at least until now. Direct products of groups abstract algebra youtube. Algebraic groups and discontinuous subgroups download book. The first author has recently shown that semisimple algebraic groups are classified up to. Subgroups of linear algebraic groups contents introduction 1 acknowledgements 4 1. These results have been applied to various problems, particularly in group theory, number theory, and algebraic geometry. Just as a subspace is a vector space within a larger vector space, a subgroup is a group within a larger group. Any subgroup of the symmetric group syms on a set s is called a permutation group or group of permutations.
Discontinuous groups in positive characteristic and. Proceedings of symposia in pure mathematics publication year 1966. Algebraic groups and discontinuous subgroups ubc math morphic form on a semisimple lie group, which encompasses the two types. Tits proved that some groups of type e8 have no others. The links to ca, gt, ft, and ag in the pdf file will work if the files are placed in. Notes on nilpotent elements in modular lie algebras 1. Seitz 1 introduction this article is a collection of notes from a series of talks given at the bernoulli center. Algebraic groups and number theory provides the first systematic exposition in mathematical literature of the junction of group theory, algebraic geometry, and number theory. Combine table 9 and the arguments used for corollary 10. Algebraic groups and discontinuous subgroups a borel, g mostow pdf. 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. Voskuil received 22 november 2001 communicated by leonard lipshitz 1.
Algebraic subgroups of lie groups dong hoon lee communicated by e. The union of subgroups a and b is a subgroup if and only if either a or b contains the other, since for example 2 and 3 are in the union of 2z and 3z but their sum 5 is not. Just as you can factor integers into prime numbers, you can break apart some groups into a direct product of simpler groups. Consider an extension of algebraic groups, that is, an exact. Let g be an algebraic group over a field k, and n a normal subgroup of g. The hasse principle, the proof for type a n here the hasse principle should be stated over a number.
Home package algebraic groups and discontinuous subgroups a borel, g mostow pdf. To do so, we rst study actions of algebraic groups. The 199293 year at the mathematical sciences research institute was devoted to complex algebraic geometry. Y kamishima properly discontinuous actions 191 we shall exhibit fiber bundle structures of discrete subgroups of algebraic groups and characterize those fibers. Featured on meta community and moderator guidelines for. Cobounded subgroups of algebraic groups over local fields. Introduction to arithmetic groups dave witte morris arxiv.
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. One often calls a 1parameter ksubgroup of g, even when ker 6 1. On extensions of algebraic groups with finite quotient. On the maximal solvable subgroups of semisimple algebraic groups. Every algebraic group g over k has a smallest normal subgroup scheme. Reductive subgroups of exceptional algebraic groups. An interesting conference article by steinberg has related results. We give a simpler proof of his result, prove that some groups of type. We will study the groups abstractly and also group the groups in some natural groups of groups decide which of the words group are technical terms. What can one say about maximal subgroups, or, more generally, the subgroup structure of simple, finite, or algebraic groups. The theory of simple algebraic groups is important in many areas of mathematics. Notation for sets and functions, basic group theory, the symmetric group, group actions, linear groups, affine groups, projective groups, finite linear groups, abelian groups, sylow theorems and applications, solvable and nilpotent groups, p groups, a second look, presentations of groups, building new groups from old. 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. Find materials for this course in the pages linked along the left.
Vaserstein department of mathematics penn state university university park, pennsylvania 16802 submitted by george p. Same definition in borel reduction theory for arithmetic groups in pspum 9 algebraic groups and discontinuous subgroups. This preprint server is intended to be a forum of the recent development of the theory of. A variety x is called complete if for every variety y the projection map x y. Algebra i basic notions of algebra kostrikin a i, shafarevich i r pdf. Linear algebraic group encyclopedia of mathematics. Algebraic groups, lie groups, and their arithmetic subgroups. Though i remembered their algorithm and table, which carry over to root systems in algebraic settings, i had completely forgotten their useful remarque ii. Cayley every group is isomorphic to a permutation group.
On the algebraic structure of discontinuous groups article pdf available in proceedings of the american mathematical society 45. Combining this for char k 2, one gets the following diagram with exact rows and columns. Parabolic subgroups, borel subgroups and solvable subgroups 12. The intersection of subgroups a and b is again a subgroup. The maximal subgroups of positive dimension in exceptional. Discontinuous groups in positive characteristic and automorphisms of mumford curves. The structure and generators of cyclic groups and subgroups theorem 5 the structure of cyclic groups, thm 7. Algebraic groups and discontinuous subgroups about this title. Pdf on the algebraic structure of discontinuous groups.
A linear algebraic group is an algebraic group that is isomorphic to an algebraic. Moreover conjugate subgroups of gare isomorphic, since the homomorphism sending h2hto ghg 1 is an isomorphism from hto ghg 1 whose inverse is the homorphism sending h02ghg 1to g h0g. As an application, it is shown that all automorphisms of the moduli. The general problem underlying this article is to give a qualitative classi. A note on quotients of real algebraic groups by arithmetic. Ga khomomorphism possibly trivial, though that case is not interesting.
They also give rise to very interesting examples of riemannian manifolds. Freydoon shahidi purdue universityeisenstein series and lfunctions november 14, 2018 6 38. Subgroups associated to a 1parameter subgroup let gbe a smooth a ne group over a eld k, and. Let r be a discrete subgroup of g, k a maximal compact subgroup of g, pa. Algebraic groups and discontinuous subgroups institut fourier. J0,4 and h0,2,4,6, where j is also a subgroup of h. Tits pindexes of semisimple algebraic groups skip garibaldi. In the algebraic group setting, the rst group consists of unipotent elements so its lie algebra consists of nilpotent elements, whereas.
1367 961 437 177 1483 1459 658 1439 879 878 802 1181 799 281 502 1222 1246 32 777 282 626 516 508 102 1486 441 1269 508 87 348 134 116 483