This book provides an introduction to Lie groups， Lie algebras， and representation theory， aimed at graduate students in mathematics and physics.Although there are already several excellent books that cover many of the same topics， this book has two distinctive features that I hope will make it a useful addition to the literature. First， it treats Lie groups （not just Lie alge bras） in a way that minimizes the amount of manifold theory needed. Thus，I neither assume a prior course on differentiable manifolds nor provide a con-densed such course in the beginning chapters. Second， this book provides a gentle introduction to the machinery of semisimple groups and Lie algebras by treating the representation theory of SU（2） and SU（3） in detail before going to the general case. This allows the reader to see roots， weights， and the Weyl group "in action" in simple cases before confronting the general theory.The standard books on Lie theory begin immediately with the general case：a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run， but it is rather abstract for a reader encountering such things for the first time. Furthermore， with this approach， one must either assume the reader is familiar with the theory of differentiable manifolds （which rules out a substantial part of one's audience） or one must spend considerable time at the beginning of the book explaining this theory （in which case， it takes a long time to get to Lie theory proper）.
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圖書(shū)目錄
Introduction Advice for the Beginner Information for the Expert Prerequisites Sources Courses Acknowledgements 0 Elementary Definitions 0.1 Rings and Ideals 0.2 Unique Factorization 0.3 Modules Ⅰ Basic Constructions 1 Roots of Commutative Algebra 1.1 Number Theory 1.2 Algebraic Curves and Fhnction Theory 1.3 Invariant Theory 1.4 The Basis Theorem 1.5 Graded Rings 1.6 Algebra and Geometry： The Nullstellensatz 1.7 Geometric Invariant Theory 1.8 Projective Varieties 1.9 Hilbert Functions and Polynomials 1.10 Free Resolutions and the Syzygy Theorem 1.11 Exercises 2 Localization 2.1 Fractions 2.2 Horn and Tensor 2.3 The Construction of Primes 2.4 Rings and Modules of Finite Length 2.5 Products of Domains 2.6 Exercises 3 Associated Primes and Primary Decomposition 3.1 Associated Primes 3.2 Prime Avoidance 3.3 Primary Decomposition 3.4 Primary Decomposition and Factoriality 3.5 Primary Decomposition in the Graded Case 3.6 Extracting Information from Primary Decomposition 3.7 Why Primary Decomposition Is Not Unique 3.8 Geometric Interpretation of Primary Decomposition 3.9 Symbolic Powers and Functions Vanishing to High Order 3.10 Exercises 4 Integral Dependence and the Nullstellensatz 4.1 The Cayley-Hamilton Theorem and Nakayamas Lemma 4.2 Normal Domains and the Normalization Process 4.3 Normalization in the Analytic Case 4.4 Primes in an Integral Extension 4.5 The Nullstellensatz 4.6 Exercises 5 Filtrations and the Artin-Rees Lemma 5.1 Associated Graded Rings and Modules 5.2 The Blowup Algebra 5.3 The Krull Intersection Theorem 5.4 The Tangent Cone 5.5 Exercises 6 Flat Families 6.1 Elementary Examples 6.2 Introduction to Tor 6.3 Criteria for Flatness 6.4 The Local Criterion for Flatness 6.5 The Rees Algebra 6.6 Exercises 7 Completions and Hensels Lemma 7.1 Examples and Definitions 7.2 The Utility of Completions 7.3 Lifting Idempotents 7.4 Cohen Structure Theory and Coefficient Fields 7.5 Basic Properties of Completion 7.6 Maps from Power Series Rings 7.7 Exercises Ⅱ Dimension Theory 8 Introduction to Dimension Theory 8.1 Axioms for Dimension 8.2 Other Characterizations of Dimension Fundamental Definitions of Dimension Theory 9.1 Dimension Zero 9.2 Exercises 10 The Principal Ideal Theorem and Systems of Parameters 10.1 Systems of Parameters and Ideals of Finite Colength 10.2 Dimension of Base and Fiber 10.3 Regular Local Rings 10.4 Exercises 11 Dimension and Codimension One 11.1 Discrete Valuation Rings 11.2 Normal Rings and Serres Criterion 11.3 Invertible Modules 11.4 Unique Factorization of Codimension-One Ideals 11.5 Divisors and Multiplicities 11.6 Multiplicity of Principal Ideals 11.7 Exercises 12 Dimension and Hilbert-Samuel Polynomials 12.1 Hilbert-Samuel Functions 12.2 Exercises 13 The Dimension of Affine Rings 13.1 Noether Normalization 13.2 The Nullstellensatz 13.3 Finiteness of the Integral Closure 13.4 Exercises 14 Elimination Theory， Generic Freeness， and the Dimension of Fibers 14.1 Elimination Theory 14.2 Generic Preeness 14.3 The Dimension of Fibers 14.4 Exercises 15 GrSbner Bases 15.1 Monomials and Terms 15.2 Monomial Orders 15.3 The Division Algorithm 15.4 Gr5bner Bases 15.5 Syzygies 15.6 History of Gr5bner Bases 15.7 A Property of Reverse Lexicographic Order 15.8 Gr5bner Bases and Flat Families 15.9 Generic Initial Ideals 15.10 Applications 15.11 Exercises 15.12 Appendix： Some Computer Algebra Projects 16 Modules of Differentials 16.1 Computation of Differentials 16.2 Differentials and the Cotangent Bundle 16.3 Colimits and Localization 16.4 Tangent Vector Fields and Infinitesimal Morphisms 16.5 Differentials and Field Extensions 16.6 Jacobian Criterion for Regularity 16.7 Smoothness and Generic Smoothness 16.8 Appendix： Another Construction of Kahler Differentials 16.9 Exercises Ⅲ Homological Methods 17 Regular Sequences and the Koszul Complex 17.1 Koszul Complexes of Lengths I and 2 17.2 Koszul Complexes in General 17.3 Building the Koszul Complex from Parts 17.4 Duality and Homotopies 17.5 The Koszul Complex and the Cotangent Bundle of Projective Space 17.6 Exercises 18 Depth， Codimension， and Cohen-Macaulay Rings 18.1 Depth 18.2 Cohen-Macaulay Rings 18.3 Proving Primeness with Serres Criterion 18.4 Flatness and Depth 18.5 Some Examples 18.6 Exercises 19 Homological Theory of Regular Local Rings 19.1 Projective Dimension and Minimal Resolutions 19.2 Global Dimension and the Syzygy Theorem 19.3 Depth and Projective Dimension： The Auslander-Buchsbaum Formula 19.4 Stably Free Modules and Factoriality of Regular Local Rings 19.5 Exercises 20 Free Resolutions and Fitting Invariants 20.1 The Uniqueness of Free Resolutions 20.2 Fitting Ideals 20.3 What Makes a Complex Exact? 20.4 The Hilbert-Burch Theorem 20.5 Castelnuovo-Mumford Regularity 20.6 Exercises 21 Duality， Canonical Modules， and Gorenstein Rings 21.1 Duality for Modules of Finite Length 21.2 Zero-Dimensional Gorenstein Rings 21.3 Canonical Modules and Gorenstein Rings in Higher Dimension 21.4 Maximal Cohen-Macaulay Modules 21.5 Modules of Finite Injective Dimension 21.6 Uniqueness and (Often) Existence 21.7 Localization and Completion of the Canonical Module 21.8 Complete Intersections and Other Gorenstein Rings 21.9 Duality for Maximal Cohen-Macaulay Modules 21.10 Linkage 21.11 Duality in the Graded Case 21.12 Exercises Appendix 1 Field Theory A1.1 Transcendence Degree A1.2 Separability A1.3 p-Bases Appendix 2 Multilinear Algebra A2.1 Introduction A2.2 Tensor Products A2.3 Symmetric and Exterior Algebras A2.4 Coalgebra Structures and Divided Powers A2.5 Schur Functors A2.6 Complexes Constructed by Multilinear Algebra Appendix 3 Homological Algebra A3.1 Introduction Part I： Resolutions and Derived Functors A3.2 Free and Projective Modules A3.3 Free and Projective Resolutions A3.4 Injective Modules and Resolutions A3.5 Basic Constructions with Complexes A3.6 Maps and Homotopies of Complexes A3.7 Exact Sequences of Complexes A3.8 The Long Exact Sequence in Homology A3.9 Derived Functors A3.10 Tor A3.11 Ext PartⅡI： From Mapping Cones to Spectral Sequences A3.12 The Mapping Cone and Double Complexes A3.13 Spectral Sequences A3.14 Derived Categories Appendix 4 A Sketch of Local Cohomology A4.1 Local Cohomology and Global Cohomology A4.2 Local Duality A4.3 Depth and Dimension Appendix 5 Category Theory A5.1 Categories， Functors， and Natural Transformations A5.2 Adjoint Functors A5.3 Representable Functors and Yonedas Lemma Appendix 6 Limits and Colimits A6.1 Colimits in the Category of Modules A6.2 Flat Modules as Colimits of Free Modules A6.3 Colimits in the Category of Commutative Algebras A6.4 Exercises Appendix 7 Where Next Hints and Solutions for Selected Exercises References Index of Notation Index
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