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).
目录
Introduction 0 Elementary defintions I Basic Constructions 1 Roots of commutative algebra 2 Localization 3 Associated Primes and Primary Decomposition 4 Integral Dependence and the nullstellensatz 5 Filtrations and the artin-rees lemma 6 Flat families 7 Completions and hensel`s lemma II Dimension theory 8 Introduction to dimension theory 9 Fundamental definitions of dimension theory 10 The principal Ideal Theorem and systems of parameters 11 Dimension and codimension one 12 Dimension and hibert samuel polynomials 13 The Dimension of affine rings 14 Elimination theory,generic freeness,and the dimension of fibers 15 Grobner Bases 16 Modules of differentials III Homological methods 17 Regular sequences and the koszul complex 18 Depth,codimension,and cohen-macaulay rings 19 Homological theory of regular local rings 20 Free resolutions and fitting invariants …… Part II:From Mapping cones to spectral Sequences Hints and solutions for selected Exercises Referneces Index of Notation Index
