Then gh is an affine variety if and only if h is reductive. Gdiagonalisable gis isomorphic to a closed subgroup of d n g g s is commutative lemma. We prove that a nonelementary subgroup hof an acylindrically hyperbolic group gis algebraic if and only if there exists a nite subgroup kof. G contains a unique maximal closed connected subgroup, denoted g0, which is normal and has nite index. Given a subgroup h and some a in g, we define the left coset ah ah. 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. Algebra 181 1996, 2669, we give a classification of all such triples g, y, v when g is a nonconnected algebraic group. Descargar algebraic groups and discontinuous subgroups. This is a linear algebraic group, which may be assumed connected after. This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds.
Linear algebraic groups and finite groups of lie type originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. On group algebras smith, martha, bulletin of the american mathematical society, 1970. Tori, unipotent and connected solvable groups 12 3. Algebraic groups and discontinuous subgroups a borel, g. Subgroups of linear algebraic groups contents introduction 1 acknowledgements 4 1.
I have chosen this order because everybody is familiar with the ring of. Strong approximation was established in the 1960s and 1970s, for semisimple simplyconnected algebraic groups over global fields. An introduction to the theory of algebraic group schemes over fields these notes have been rewritten and published 2017. In weak approximation theorems the product is over a finite set of places s, while in strong approximation theorems the product is over all but a finite set of places history. On irreducible subgroups of simple algebraic groups 1261 where the wi are pairwise orthogonal subspaces of w. Our construction is based on that of subgroups of g acting nonseparably on the unipotent radical of a proper parabolic. Irreducible subgroups of classical algebraic groups. Reductive subgroups of exceptional algebraic groups. Thanks for contributing an answer to mathematics stack exchange. Abstract algebra number theory, group theory, galois. Rapinchuk we survey results on strong approximation in algebraic groups, considering in detail the classical form of strong approximation as well as more recent results on strong approximation for arbitrary zariskidense subgroups. A computational introduction to number theory and algebra victor shoups. Jacobson abstract a subgroup of a group gis called algebraic if it can be expressed as a nite union of solution sets to systems of equations. Algebraic subgroups of acylindrically hyperbolic groups b.
Langlands suppose pand p0are two standard cuspidal subgroups with the split components a and a0 respectively. They give conditions for the group gk to be dense in a restricted direct product of groups of the form gk s for k s a completion of k at the place s. As a consequence we deduce that ifg has more than one simple factor. In this paper, we present the first examples of nonabelian nongcompletely reducible k subgroups of g which are gcompletely reducible over k. Sep 04, 2006 borel, mostow algebraic groups and discontinuous subgroups. It is shown in this paper that ifg is the group ofkpoints of a semisimple algebraic groupg over a local fieldk of positive characteristic such that all itsksimple factors are ofkrank 1 and. Algebraic groups and discontinuous subgroups a borel, g mostow pdf. It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of lie type are known. Subgroup structure and representations of finite and. They are maximal parabolics, maximalrank subgroups, 22 d 4. Complete reducibility of subgroups of reductive algebraic. In his work on the approximation of continuous lie groups by discrete subgroups, kuranishi 29 proved in the 1950s that every. Weakly commensurable sarithmetic subgroups in almost simple algebraic groups of types b and c 27 p. Algebraic groups an algebraic group over k is an a ne variety g km with a compatible group structure.
G u are closed subgroups of g, and g s g u gis an isomorphism of a ne algebraic groups. It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of lie type. Maximal subgroups of exceptional algebraic groups the maximal subgroups m of positive dimension in exceptional algebraic groups have been completely classi ed by liebeck and seitz. Contributions to function theory, tata institute of fundamental research, bombay 1960. In this paper we study the collection of connected girreducible subgroups of semisimple algebraic groups g. The maximal subgroups of positive dimension in exceptional.
Parabolic subgroups and flag varieties so far, weve been studying klein geometry, in particular projective geometry with symmetry group gln. These two conditions can be combined into one equivalent condition. The institute was held at the university of colorado in boulder from july s to august 6, 1965, and was financed by the national science foundation and the office of naval research. Algebraic groups and discontinuous subgroups author. Simply connected algebraic groups and reductive subgroups. The sequel course \linear algebraic groups ii treats the boreltits structure theory of reductive groups. 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. First appeared in algebraic groups and discontinuous subgroups, ams, proc. These groups are algebraically simple no nontrivial normal subgroups. Borel, mostow algebraic groups and discontinuous subgroups.
A note on quotients of real algebraic groups by arithmetic subgroups. But avoid asking for help, clarification, or responding to other answers. Algebraic groups and discontinuous subgroups about this title. 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 subset h of the group g is a subgroup of g if and only if it is nonempty and closed under products and inverses. Let g be a connected reductive complex algebraic group, h an algebraic subgroup. Furthermore, every element of g is contained in precisely one left coset of h. Jordan decompositions are preserved by homomorphisms of algebraic groups. Properly discontinuous actions of subgroups in amenable.
This will show that every nontrivial proper connected reductive fsubgroup of g is a maximal torus. Algebraic groups and discontinuous subgroups a borel. On irreducible subgroups of simple algebraic groups. The results for number fields are due to kneser 1966 and platonov 1969. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classi. Raghunathan 1, 2 inventiones mathematicae volume 4, pages 318 335 1968 cite this article. Indeed, let h be a nontrivial proper freductive subgroup of g. Theorem 1 let gbe a connected semisimple algebraic group, and let abe a connected girreducible subgroup of g. Proceedings of symposia in pure mathematics publication year 1966. In this paper and in overgroups of irreducible linear groups, i j. 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. Algebraalgebraic groups and discontinuous subgroups. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups.
Strong approximation for algebraic groups 273 and in particular, x. This book is an outgrowth of the twelfth summer mathematical institute of the american mathematical society, which was devoted to algebraic groups and discontinuous subgroups. The structure and generators of cyclic groups and subgroups theorem 5 the structure of cyclic groups, thm 7. A note on quotients of real algebraic groups by arithmetic. On discontinuous groups in higher dimensional symmetric spaces. Large collection of advanced mathematics ebooks books. Large collection of advanced mathematics ebooks books pics.
Standard parabolic subgroups and the levi decomposition 39. 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. We say that a subgroup h of g normalizes such a decomposition if it. Algebraic groups lecture notes uw courses web server. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. A note on quotients of real algebraic groups by arithmetic subgroups m. I will be using the definitions and conventions set out in malle and testermans linear algebraic groups and finite groups of lie type. But according to chevalleys structure theorem any algebraic group is an extension of an abelian variety by a linear algebraic group. Currently this section contains no detailed description for the page, will update this page soon.
Theorem structure of commutative linear algebraic groups g. In this paper, we present the first examples of nonabelian nongcompletely reducible ksubgroups of g which are gcompletely reducible over k. Borovik, borovik mirrors and reflections the geometry of finite reflection groups. We study the question of which torsion subgroups of commutative algebraic groups over. Preface these are the notes of the secondyear course algebraic structures i at queen.
Let g be a ksplit simple algebraic group of type e 6 or g 2 defined over k. Algebraic groups with few subgroups page 3 of 26 1 g does not contain any fde. Your argument against subgroups of small index looks most efficient. Jun 14, 2005 project euclid mathematics and statistics online. This course introduces classical and new results on the algebraic structure of the identity component of the di. Algebraic groups and discontinuous subgroups download book.
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