新書推薦：
《
《阳明先生年谱》明刻本合集对照
》
售價：HK$
140.8
《
女人30+，养气血、调脾胃、防衰老（北京卫视《养生堂》《我是大医生》特邀中医养生专家第七批名老中医沈
》
售價：HK$
61.6
《
征服新世界：西班牙全球帝国之路，1493—1898
》
售價：HK$
96.8
《
不纠结了 摆脱选择焦虑的极简法则 高敏感、优柔寡断、总陷入选择内耗的 纠结星人 自救指南
》
售價：HK$
54.8
《
当战争来敲门（牛津二战史，耶鲁、剑桥大学教授推荐）
》
售價：HK$
96.8
《
初老的身体：更年期新规则指南（第2版）
》
售價：HK$
140.8
《
摘星星的人
》
售價：HK$
47.1
《
名侦探的守则（东野圭吾：反套路！玩梗诡计，吐槽套路，爆笑推理！“你怎么看出我男扮女装的？”）
》
售價：HK$
64.9
|
| 內容簡介： |
|
The first part of this book on Discrete Subgroups of Lie Groups is written by E.B. Vinberg. V.V. Gorbatsevich， and O.V. Shvartsman. Various types of discrete subgroups of Lie groups arise in the theory of functions of complex variables. arithmetic. geometry. and crystallography. S？nce the foundation of their general theory in the 50-60s of this century， considerable and in many respects exhaustive results were obtained. This development is reflected in this survey. Both semisimple and general Lie groups are considered. Partll on Cohomologies of Lie Groups and Lie Algebras is written by B.L. Feigin and D.B. Fuchs. It contains different definitions of cohomologies of Lie groups and (both finite-dimensional and some infinitedimensional) Lie algebras， the main methods of their calculation， and the results of these calculations.The book can be useful as a reference and research guide to graduate students and researchers in different areas of mathematics and theoretical physics.
|
| 目錄：
|
|
Contents I. Discrete Subgroups of Lie Groups E. B. Vinberg， Y. Y. Gorbatsevich and O.Y. Shvartsman 1II. Cohomologies of Lie Groups and Lie Algebras B. L. Feigin and D. B. Fuchs 125 Author Index 217 Subject Index 221 I. Discrete Subgroups of Lie Groups E. B. Vinberg ， V V Gorbatsevich and O.V Shvartsman Contents Chapter 0. Introduction 5 Chapter 1. Discrete Subgroups of Locally Compact Topological Groups 7 1. The Simplest Properties of Lattices 7 1.1. Definition of a Discrete Subgroup. Examples 7 1.2. Commensurability and Reducibility of Lattices 10 2. Discrete Groups of Transformations 12 2.1. Basic Definitions and Examples 12 2.2. Covering Sets and Fu ndamental Domains of a Discrete Group of Transformations 15 3. Group-Theoretical Properties of Lattices in Lie Groups 17 3.1. Finite Presentability of Lattices 17 3.2. A Theorem of Selberg and Some of Its Consequences 18 3.3. The Property (T) 18 4. Intersection of Discrete Subgroups with Closed Subgroups 20 4.1. T-Closedness of Subgroups 20 4.2. Subgroups with Good r-Heredity 22 4.3. Quotient Groups with Good r-Heredity 23 4.4. T-closure 24 5. The Space of Lattices of a Locally Compact Group 25 5.1. Chabauty‘s Topology 25 5.2. Minkowski’s Lemma 25 5.3. Mahler‘s Criterion 26 6. Rigidity of Discrete Subgroups of Lie Groups 27 6.1. Space of Homomorphisms and Deformations 27 6.2. Rigidity and Cohomology 28 6.3. Deformation of Uniform Subgroups 30 7. Arithmetic Subgroups of Lie Groups 31 7.1. Definition of an Arithmetic Subgroup 31 7.2. When Are Arithmetic Groups Lattices (Uniform Lattices) 32 7.3. The Theorem of Borel and Harish-Chandra and the Theorem of Godement 34 7 4. Definition of an Arithmetic Subgroup of a Lie Group 35 8. The Borel Density Theorem 37 8.1. The Property (S) 37 8.2. Proof of the Density Theorem 37 Chapter 2. Lattices in Solvable Lie Groups 39 1. Discrete Subgroups in Abelian Lie Groups 39 1.1. Historical Remarks 39 1.2. Structurc of Discrete Subgroups in Simply-Connected Abelian Lic Groups 39 1.3. Structure of Discrcte Subgroups in Arbitrary Connected Abelian Groups 40 1.4. Use of thc Languagc of thc Thcory of AIgebraic Groups 41 1.5. Extendability of Latticc Homomorphisms 41 2. Lattices in Nilpotent Lic Groups 42 2.1. Introductory Remarks and Examplcs 42 2.2. Structurc of Latticcs in Nilpotcnt Lic Groups 43 2.3. Latticc Homomorphisms in Nilpotcnt Lie Groups 45 2 4. Existcncc of Latticcs in Nilpotcnt Lic Groups and Thcir Classification 46 2.5. Lattices and Lattice Subgroups in Nilpotent Lie Groups 47 3. Lattices in Arbitrary Solvable Lie Groups 48 3.1. Examples of Lattices in Solvable Lie Groups of Low Dimension 48 3.2. Topology of Solvmanifolds of Type R/T 49 3.3. Some General Properties of Lattices in Solvable Lie Groups.50 3.4. Mostow’s Structure Theorem 51 3.5. Wang Groups 51 3.6. Splitting of Solvable Lie Groups 52 3.7. Criteria for the Existence of a Lattice in a Simply-Connected Solvable Lie Group 55 3.8. Wang Splitting and its Applications 55 3.9. Algebraic Splitting and its Applications 58 3.10. Linear Representability of Lattices 60 4. Deformations and Cohomology of Lattices in Solvable Lie Groups 61 4.1. Description of Deformations of Lattices in Simply-Connected Lie Groups 61 4.2. On the Cohomology of Lattices in Solvable Lie Groups 63 5. Lattices in Special Classes of Solvable Lie Groups 64 5.1. Lattices in Solvable Lie Groups of Type (I) 64 5.2. Lattices in Lie Groups of Type (R) 65 5.3. Lattices in Lie Groups of Type (E) 65 5.4. Lattices in Complex Solvable Lie Groups 66 5.5. Solvable Lie Groups of Small Dimension， Having Lattices 66 Chapter 3. Lattices in Semisimple Lie Groups 67 1. General Information 68 1.1. Red ucibility of Lattices 68 1.2. The Density Theorem 68 2. Reduction Theory 69 2.1. Geometrical Language. Construction of a Reduced Basis 69 2.2. Proof of Mahler’s Criterion 72 2.3. The Siegel Domain 72 3. The Theorem of Borel and Harish-Chandra(Continuation) 75 3.1. The Case of a Torus 75 3.2. The Semisimple Case (Siegel Domains) 77 3.3. Proof of Godement‘s Theorem in the Semisimple Case 78 4. Criteria for Uniformity of Lattices. Covolumes of Lattices 79 4.1. Unipotent Elements in Lattices 79 4.2. Covolumes of Lattices in Semisimple Lie Groups 80 5. Strong Rigidity of Lattices in Semisimple Lie Groups 82 5.1. A Theorem on Strong Rigidity 82 5.2. Satake Compactifications of Symmetric Spaces 83 5.3. Plan of the Proof of Mostow’s Theorem 85 6. Arithmetic Subgroups 86 6.1. The Field Restriction Functor E 87 6.2. Construction of Arithmetic Lattices 90 6.3. Maximal Arithmetic Subgroups 92 6 4. The Commensurator 94 6.5. Normal Subgroups of
|
| 內容試閱：
|
|
Chapter 0. Introduction The foundations of the theory of discrete subgroups of Lie groups were laid down in the fifties and sixties of this century in the papers of A. 1. Maltsev， G. Mostow， L. Auslander， and a number of other mathematicians. The way had been prepared by other investigations into special classes of discrete groups， owing their origins to arithmetic， geometry， the theory of functions ， and to physics. The first nontrivial discrete subgroup-the subgroup SL 2 (Z) of the group SL 2 (R) ， later called the modular Klein group-was considered in essence by Lagrange and Gauss in their investigations into the arithmetic of quadratic forms in two variables. Its natural generalization was the subgroup SLn(Z) of the group S Ln (R) . The investigation of this group， as a discrete group of transformations of the space of positive definite quadratic forms in n variables， constituted the objective of reduction theory， worked out by A. N. Korkin and E. 1. Zolotarev， Hermite， Minkowski， and others in the second half ot the nineteenth century， and at the beginning of this. A number of other arithmetically defined discrete subgroups of the classical Lie groups-the groups of units of rational quadratic forms ， the groups of units of simple algebras over the field Q of rationals ， the group of integer symplectic matrices-were studied in the first half ofthis century by B. A. Venkov ， H. Weyl ， C. L. Siegel and others. In the theory of functions of a complex variable the integration of algebraic functions， and ， more generally， the solution of linear differential equations with algebraic coefficients ， led to the consideration of certain special functions ， later called automorphic ， invariant relative to various discrete subgroups of the group SL2 (R) ， operating in the upper halfplane by fractional-linear transformations. Some of the discrete subgroups of the group SL2 (R) arising in this way were studied in the middle of the nineteenth century by Hermite， Dedekind， and Fuchs. Among these was the group SL2 (Z) ， but represented in a form different from that of Lagrange and Gauss. A wide class of such groups， including the group SL2 (Z) and some subgroups of SL2 (R) commensurable with it ， were studied by Klein. Almost simultaneously， Poincaré in 1881-1882 gave a geometrical description of all the discrete groups of fractional-linear transformations of the upper halfplane， called by him the Fu chsian groups. In the first half of this century a number of separate classes of meromorphic functions of several variables were considered. These functions were connected with arithmetically defined discrete subgroups of the groups (SL2 (R))k (the modular functions of Hilbert) ， SP2n(R) (the modular functions of Siegel) ， and of other semisimple Lie groups. In crystallography， beginning at the end of the last century， symmetry groups of crystal structures were studied. These are discrete subgroups of the group of motions of three-dimensional Euclidean space. E. S. Fedorov and A. Schoenflies obtained the classification of such groups. Analogous groups of motions of n-dimensional Euclidean space were studied in 1911 by Bieberbach. Another branch was the study of discrete subgroups of solvable Lie groups， in particular abelian and nilpotent. The first result on such groups， equivalent to the description of discrete subgroups in JR2 ， was obtained by Jacobi in the first half of the last century， in the course of describing the periods of meromorphic functions. In the present work we have tried to systematize all the basic results on the theory of discrete subgroups of Lie groups. Most of it has the character of a survey. But in those cases when there is a short proof， and in particular when no short proof has yet been published ， we present one. Apart from the original papers， our basic sources were: the monographs of Raghunathan (1972) ， Mostow (1973) and Zimmer (1984) ， the surveys of H.-C Wang (1972) ， Mostow (1978A) ， Auslander (1973) ， Margulis (1974) ， and finally the notes for specialized courses given by the first-named author at Moscow State University. A more detailed exposition of the theory of discrete subgroups of motions in spaces of constant curvature is given in the paper of Vinberg and Shvartsman (1988) in volume 29 ofthis Encyclopaedia， which deals particularly spaces of constant curvature. We have adopted the following notations and conventions. N denotes the natural numbers， Z the integer numbers ， Q the field of the rationals ， JR that of the reals ， and C that of the complex numbers. If a Lie group is denoted by a capital latin letter， such as H ， then its tangent Lie algebra will be denoted by the corresponding small gothic letter ， in the above case . A connected component of a topological group G will be denoted by GO. The universal covering of G will be denoted by G. If H is a subset of G ， we will denote by H its closure in the topology of
|
|
購物車/收銀台(0) | 在線留言板
| 付款方式
| 運費計算
| 聯絡我們
| 幫助中心 |
加入書簽
HOME
新書上架
暢銷書架
