Discrete Finite And Lie Groups

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Discrete Subgroups Of Semisimple Lie Groups

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Author : Gregori A. Margulis
language : en
Publisher: Springer Science & Business Media
Release Date : 1991-02-15

Discrete Subgroups Of Semisimple Lie Groups written by Gregori A. Margulis and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-02-15 with Mathematics categories.

Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. Since the notion of "Lie group" is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e.g. classification results on abstract homomorphisms of semi-simple algebraic groups over global fields. The treatise of course contains a presentation of the author's fundamental rigidity and arithmeticity theorems. The work in this monograph requires the language and basic results from fields such as algebraic groups, ergodic theory, the theory of unitary representatons, and the theory of amenable groups. The author develops the necessary material from these subjects; so that, while the book is of obvious importance for researchers working in related areas, it is essentially self-contained and therefore is also of great interest for advanced students.

Discrete Subgroups Of Lie Groups

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Author : Madabusi S. Raghunathan
language : en
Publisher: Springer
Release Date : 2012-11-09

Discrete Subgroups Of Lie Groups written by Madabusi S. Raghunathan and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-11-09 with Mathematics categories.

This book originated from a course of lectures given at Yale University during 1968-69 and a more elaborate one, the next year, at the Tata Institute of Fundamental Research. Its aim is to present a detailed ac count of some of the recent work on the geometric aspects of the theory of discrete subgroups of Lie groups. Our interest, by and large, is in a special class of discrete subgroups of Lie groups, viz., lattices (by a lattice in a locally compact group G, we mean a discrete subgroup H such that the homogeneous space GJ H carries a finite G-invariant measure). It is assumed that the reader has considerable familiarity with Lie groups and algebraic groups. However most of the results used frequently in the book are summarised in "Preliminaries"; this chapter, it is hoped, will be useful as a reference. We now briefly outline the contents of the book. Chapter I deals with results of a general nature on lattices in locally compact groups. The second chapter is an account of the fairly complete study of lattices in nilpotent Lie groups carried out by Ma1cev. Chapters III and IV are devoted to lattices in solvable Lie groups; most of the theorems here are due to Mostow. In Chapter V we prove a density theorem due to Borel: this is the first important result on lattices in semisimple Lie groups.

An Introduction To Lie Groups And Lie Algebras

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Author : Alexander Kirillov, Jr
language : en
Publisher: Cambridge University Press
Release Date : 2017-06-30

An Introduction To Lie Groups And Lie Algebras written by Alexander Kirillov, Jr and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-30 with Mathematics categories.

This classic graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Lie theory, in its own right, has become regarded as a classical branch of mathematics. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras.

Discrete Finite And Lie Groups

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Author : Pietro Giuseppe Fré
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2023-08-07

Discrete Finite And Lie Groups written by Pietro Giuseppe Fré and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-07 with Mathematics categories.

In a self contained and exhaustive work the author covers Group Theory in its multifaceted aspects, treating its conceptual foundations in a proper logical order. First discrete and finite group theory, that includes the entire chemical-physical field of crystallography is developed self consistently, followed by the structural theory of Lie Algebras with a complete exposition of the roots and Dynkin diagrams lore. A primary on Fibre-Bundles, Connections and Gauge fields, Riemannian Geometry and the theory of Homogeneous Spaces G/H is also included and systematically developed. https://petrusfremathandlit.net

Lie Groups

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Author : J.J. Duistermaat
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Lie Groups written by J.J. Duistermaat and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

This book is devoted to an exposition of the theory of finite-dimensional Lie groups and Lie algebras, which is a beautiful and central topic in modern mathematics. At the end of the nineteenth century this theory came to life in the works of Sophus Lie. It had its origins in Lie's idea of applying Galois theory to differential equations and in Klein's "Erlanger Programm" of treat ing symmetry groups as the fundamental objects in geometry. Lie's approach to many problems of analysis and geometry was mainly local, that is, valid in local coordinate systems only. At the beginning of the twentieth century E. Cartan and Weyl began a systematic treatment of the global aspects of Lie's theory. Since then this theory has ramified tremendously and now, as the twentieth century is coming to a close, its concepts and methods pervade mathematics and theoretical physics. Despite the plethora of books devoted to Lie groups and Lie algebras we feel there is justification for a text that puts emphasis on Lie's principal idea, namely, geometry treated by a blend of algebra and analysis. Lie groups are geometrical objects whose structure can be described conveniently in terms of group actions and fiber bundles. Therefore our point of view is mainly differential geometrical. We have made no attempt to discuss systematically the theory of infinite-dimensional Lie groups and Lie algebras, which is cur rently an active area of research. We now give a short description of the contents of each chapter.

Harmonic Analysis And Representations Of Semisimple Lie Groups

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Author : J.A. Wolf
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Harmonic Analysis And Representations Of Semisimple Lie Groups written by J.A. Wolf and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.

This book presents the text of the lectures which were given at the NATO Advanced Study Institute on Representations of Lie groups and Harmonic Analysis which was held in Liege from September 5 to September 17, 1977. The general aim of this Summer School was to give a coordinated intro duction to the theory of representations of semisimple Lie groups and to non-commutative harmonic analysis on these groups, together with some glance at physical applications and at the related subject of random walks. As will appear to the reader, the order of the papers - which follows relatively closely the order of the lectures which were actually give- follows a logical pattern. The two first papers are introductory: the one by R. Blattner describes in a very progressive way a path going from standard Fourier analysis on IR" to non-commutative harmonic analysis on a locally compact group; the paper by J. Wolf describes the structure of semisimple Lie groups, the finite-dimensional representations of these groups and introduces basic facts about infinite-dimensional unitary representations. Two of the editors want to thank particularly these two lecturers who were very careful to pave the way for the later lectures. Both these chapters give also very useful guidelines to the relevant literature.

Lie Algebras Of Finite And Affine Type

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Author : Roger William Carter
language : en
Publisher: Cambridge University Press
Release Date : 2005-10-27

Lie Algebras Of Finite And Affine Type written by Roger William Carter and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-10-27 with Mathematics categories.

This book provides a thorough but relaxed mathematical treatment of Lie algebras.

Geometry Analysis And Topology Of Discrete Groups

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Author : Lizhen Ji
language : en
Publisher:
Release Date : 2008

Geometry Analysis And Topology Of Discrete Groups written by Lizhen Ji and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

Presents 15 papers treating discrete groups as they occur in areas such as algebra, analysis, geometry, number theory and topology. This work helps graduate students and researchers to understand the structures and applications of discrete subgroups of Lie groups and locally symmetric spaces.

Arithmetic Groups And Their Generalizations

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Author : Lizhen Ji
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Arithmetic Groups And Their Generalizations written by Lizhen Ji and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as $\mathbf{Z}$ or $\textrm{SL}(n, \mathbf{Z})$. Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known. Indeed, arithmetic groups admit many natural and important generalizations. The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry. It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics. Titles in this series are co-published with International Press, Cambridge, MA.Table of Contents: Introduction; General comments on references; Examples of basic arithmetic groups; General arithmetic subgroups and locally symmetric spaces; Discrete subgroups of Lie groups and arithmeticity of lattices in Lie groups; Different completions of $\mathbb{Q}$ and $S$-arithmetic groups over number fields; Global fields and $S$-arithmetic groups over function fields; Finiteness properties of arithmetic and $S$-arithmetic groups; Symmetric spaces, Bruhat-Tits buildings and their arithmetic quotients; Compactifications of locally symmetric spaces; Rigidity of locally symmetric spaces; Automorphic forms and automorphic representations for general arithmetic groups; Cohomology of arithmetic groups; $K$-groups of rings of integers and $K$-groups of group rings; Locally homogeneous manifolds and period domains; Non-cofinite discrete groups, geometrically finite groups; Large scale geometry of discrete groups; Tree lattices; Hyperbolic groups; Mapping class groups and outer automorphism groups of free groups; Outer automorphism group of free groups and the outer spaces; References; Index. Review from Mathematical Reviews: ...the author deserves credit for having done the tremendous job of encompassing every aspect of arithmetic groups visible in today's mathematics in a systematic manner; the book should be an important guide for some time to come.(AMSIP/43.

P Adic Lie Groups

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Author : Peter Schneider
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-06-11

P Adic Lie Groups written by Peter Schneider and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-06-11 with Mathematics categories.

Manifolds over complete nonarchimedean fields together with notions like tangent spaces and vector fields form a convenient geometric language to express the basic formalism of p-adic analysis. The volume starts with a self-contained and detailed introduction to this language. This includes the discussion of spaces of locally analytic functions as topological vector spaces, important for applications in representation theory. The author then sets up the analytic foundations of the theory of p-adic Lie groups and develops the relation between p-adic Lie groups and their Lie algebras. The second part of the book contains, for the first time in a textbook, a detailed exposition of Lazard's algebraic approach to compact p-adic Lie groups, via his notion of a p-valuation, together with its application to the structure of completed group rings.