Consider an extension of algebraic groups, that is, an exact. Jun 14, 2005 on algebraic groups and discontinuous groups. Parabolic subgroups, borel subgroups and solvable subgroups 12. We switch to the dual category of finitely generated abelian groups with a continuous action. In the algebraic group setting, the rst group consists of unipotent elements so its lie algebra consists of nilpotent elements, whereas. In general, subgroups of cyclic groups are also cyclic. Just as a subspace is a vector space within a larger vector space, a subgroup is a group within a larger group. Cobounded subgroups of algebraic groups over local fields.
Tits proved that some groups of type e8 have no others. Arithmetic subgroups of algebraic groups by armand borel and harishchandra communicated by deane montgomery, july 22, 1961. Algebraic groups and discontinuous subgroups a borel, g mostow pdf. 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. Though i remembered their algorithm and table, which carry over to root systems in algebraic settings, i had completely forgotten their useful remarque ii. Algebraic groups and discontinuous subgroups group. Arithmetic subgroups of algebraic groups by armand borel and harishchandra received october 18, 1961 table of contents. The cayley table for h is the topleft quadrant of the cayley table for g. Barker abstract arithmetic subgroups of simple isotropic algebraic groups are described as subgroups full of root elements.
These results have been applied to various problems, particularly in group theory, number theory, and algebraic geometry. On the geometry of stable discontinuous subgroups acting on threadlike homogeneous spaces article in mathematical notes 895. We record some further nice properties of integral extensions. A subgroup h of a group gis said to be a normal subgroup of gif ghg 1 2hfor all h2hand g2g. A linear algebraic group is an algebraic group that is isomorphic to an algebraic subgroup of a general linear group. The structure and generators of cyclic groups and subgroups theorem 5 the structure of cyclic groups, thm 7. This preprint server is intended to be a forum of the recent development of the theory of. Reductive subgroups of exceptional algebraic groups. Let r be a discrete subgroup of g, k a maximal compact subgroup of g, pa. Geometric group theory preliminary version under revision. The links to ca, gt, ft, and ag in the pdf file will work if the files are placed in. The theory of simple algebraic groups is important in many areas of mathematics. Here are some fundamental facts about complete varieties. Discontinuous groups in positive characteristic and automorphisms of mumford curves.
It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of lie type are known. Notes on nilpotent elements in modular lie algebras 1. We do prove weaker results for all algebraically closed. Algebraic groups and discontinuous subgroups a borel, g. We give a simpler proof of his result, prove that some groups of type. This volume collects articles that arose from this event, which took place at a time when algebraic geometry was undergoing a major change. Then, in section 6, we combine this characterization of. Ga khomomorphism possibly trivial, though that case is not interesting. Matsumoto, subgroups of finite index in certain arithmeticgroups, in sym posium on algebraic groups and discontinuous subgroups, amer. 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. One often calls a 1parameter ksubgroup of g, even when ker 6 1.
On the geometry of stable discontinuous subgroups acting. We study the question of which torsion subgroups of commutative algebraic groups over. A computational introduction to number theory and algebra victor shoups. Descargar algebraic groups and discontinuous subgroups. Algebraic groups and discontinuous subgroups download book. Algebraic groups and discontinuous subgroups author. Mostow algebraic groups and discontinuous subgroups, proc. A linear algebraic group is an algebraic group that is isomorphic to an algebraic. Algebraic groups and discontinuous subgroups institut fourier. Currently this section contains no detailed description for the page, will update this page soon. In section 4, we study borel subgroups of arbitrary linear algebraic groups.
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. Descargar algebraic groups and number theory en pdf. 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. A group action on a finitedimensional vector space is called a representation of the group. Algebraic groups and number theory provides the first systematic exposition in mathematical literature of the junction of group theory, algebraic geometry, and number theory. Tits pindexes of semisimple algebraic groups skip garibaldi. Browse other questions tagged abstract algebra group theory finite groups sylowtheory p groups or ask your own question. 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. Let r be a discrete subgroup of g, k a maximal compact subgroup. Discontinuous subgroups of pgl connecting repositories. Let g be an algebraic group over a field k, and n a normal subgroup of g. A variety x is called complete if for every variety y the projection map x y. Tori, unipotent and connected solvable groups 12 3.
On irreducible subgroups of simple algebraic groups 1261 where the wi are pairwise orthogonal subspaces of w. On irreducible subgroups of simple algebraic groups. Cayley every group is isomorphic to a permutation group. Spinors and essential dimension compositio mathematica. This milestone work on the arithmetic theory of linear algebraic groups is now available in english for the first time. 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. Algebraic groups play much the same role for algebraists as lie groups play for analysts. Algebraic subgroups of lie groups dong hoon lee communicated by e. The general problem underlying this article is to give a qualitative classi. 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.
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 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. Subgroup structure and representations of finite and. We also explain how to make a quotient ghinto a linear algebraic group when his a closed normal subgroup of g. Maximal subgroups of exceptional groups of lie type and.
The exposition of the topic is built on a synthesis of methods from algebraic geometry, number theory. On the properly discontinuous subgroups of affine motions. This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Compact subgroups of linear algebraic groups by richard pink fakultat fu. Algebraic groups and discontinuous subgroups about this title. Algebraic groups, lie groups, and their arithmetic subgroups. Classification of algebraic simple groups, algebraic groups and discontinuous subgroups, proc.
Algebraic groups and discontinuous subgroups free ebook download as pdf file. 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. Find materials for this course in the pages linked along the left. These groups are algebraic groups, and we shall look only at representations g glv that are homomorphisms of algebraic groups. Simply connected algebraic groups and reductive subgroups of. 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.
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. A complex algebraic group g is in this note a subgroup of gln, c. What can one say about maximal subgroups, or, more generally, the subgroup structure of simple, finite, or algebraic groups. On combining these statements, we see that the ci form a complete set of orthogonal.
The maximal subgroups of positive dimension in exceptional. The hasse principle, the proof for type a n here the hasse principle should be stated over a number. We combine his methods with the recent computations of torsion indexes. Functional analysis local field algebraic group these keywords were added by machine and not by the authors. 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. An interesting conference article by steinberg has related results. Voskuil received 22 november 2001 communicated by leonard lipshitz 1.
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. The attendees ranged from people who have never studied algebraic groups to experts. Strongly dense free subgroups of semisimple algebraic groups 3 are contained in some. Representations of finite groups pdf 75p download book. J0,4 and h0,2,4,6, where j is also a subgroup of h. Introduction to arithmetic groups dave witte morris arxiv. 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.
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. We denote here the discrete invariants associated to algebras with involution. Pdf on the algebraic structure of discontinuous groups. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. Home package algebraic groups and discontinuous subgroups a borel, g mostow pdf. 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. A note on generators for arithmetic subgroups of algebraic groups. Linear algebraic group encyclopedia of mathematics.
Every algebraic group g over k has a smallest normal subgroup scheme. Algebraic groups and discontinuous subgroups, proc. As an application, it is shown that all automorphisms of the moduli. On extensions of algebraic groups with finite quotient. The direct product is a way to combine two groups into a new, larger group. Direct products of groups abstract algebra youtube. Craven birmingham maximal subgroups 4th march, 2014 9 20. On arithmetic subgroups of simple algebraic groups l. 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. Algebraic groups and discontinuous subgroups ubc math morphic form on a semisimple lie group, which encompasses the two types. Proceedings of symposia in pure mathematics publication year 1966. The group of rigid motions of a regular ngon is called the nth dihedral group, denoted by d n.
Combining this for char k 2, one gets the following diagram with exact rows and columns. The intersection of subgroups a and b is again a subgroup. Any subgroup of the symmetric group syms on a set s is called a permutation group or group of permutations. Just as you can factor integers into prime numbers, you can break apart some groups into a direct product of simpler groups. 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. Then gis a group, and we can write out its multiplication table cayley table. This process is experimental and the keywords may be. A note on quotients of real algebraic groups by arithmetic subgroups. A note on quotients of real algebraic groups by arithmetic. Another example is the union of the xaxis and the yaxis in the plane with the addition operation.
Same definition in borel reduction theory for arithmetic groups in pspum 9 algebraic groups and discontinuous subgroups. Freydoon shahidi purdue universityeisenstein series and lfunctions november 14, 2018 6 38. A quasiprojective variety is complete if and only if it is projective. Algebraic subgroups of lie groups heldermannverlag. On the algebraic structure of discontinuous groups article pdf available in proceedings of the american mathematical society 45. We can combine theorem 19 i, corollary 5 ii and theorem 20 to show. Subgroups of linear algebraic groups contents introduction 1 acknowledgements 4 1. On the maximal solvable subgroups of semisimple algebraic groups. Algebra i basic notions of algebra kostrikin a i, shafarevich i r pdf. Combine table 9 and the arguments used for corollary 10. 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. Featured on meta community and moderator guidelines for. Properly discontinuous actions of subgroups in amenable.
On arithmetic subgroups of simple algebraic groups. Discontinuous groups in positive characteristic and. Subgroups associated to a 1parameter subgroup let gbe a smooth a ne group over a eld k, and. The first author has recently shown that semisimple algebraic groups are classified up to. 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. Seitz 1 introduction this article is a collection of notes from a series of talks given at the bernoulli center. 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. Y kamishima properly discontinuous actions 191 we shall exhibit fiber bundle structures of discrete subgroups of algebraic groups and characterize those fibers.