李群、李代数和表示论

出版时间：2007-10  出版社：世界图书出版公司  作者：Brian C. Hall  页数：351  
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内容概要

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).

书籍目录

Part I General Theory  Matrix Lie Groups  1.1  Definition of a Matrix Lie Group      1.1.1  Counterexa~ples  1.2  Examples of Matrix Lie Groups      1.2.1  The general linear groups GL(n;R) and GL(n;C)      1.2.2 The special linear groups SL(n; R) and SL(n; C)      1.2.3  The orthogonal and special orthogonal groups, O(n) and SO(n)    1.2.4  The unitary and special unitary groups, U(n) and SU(n)    1.2.5 The complex orthogonal groups, O(n; C) and SO(n; C)     1.2.6  The generalized orthogonal and Lorentz groups    1.2.7 The symplectic groups Sp(n; R), Sp(n;C), and $p(n)     1.2.8  The Heisenberg group H  .    1.2.9  The groups R, C*, S1,  and Rn    1.2.10 The Euclidean and Poincaxd groups E(n) and P(n; 1)  1.3  Compactness    1.3.1  Examples of compact groups    1.3.2  Examples of noncompa groups  1.4  Connectedness  1.5  Simple Connectedness    1.6  Homomorpliisms and Isomorphisms    1.6.1 Example: SU(2) and S0(3)   1.7 The Polar Decomposition for S[(n; R) and SL(n; C)   1.8  Lie Groups   1.9  Exercises2   Lie Algebras and the Exponential Mapping   2.1  The Matrix Exponential   2.2  Computing the Exponential of a Matrix     2.2.1  Case 1: X is diagonalizable     2.2.2  Case 2: X is nilpotent     2.2.3  Case 3: X arbitrary  2.3  The Matrix Logarithm  2.4  Further Properties of the Matrix Exponential  2.5  The Lie Algebra of a Matrix Lie Group    2.5.1  Physicists' Convention    2.5.2  The general linear groups    2.5.3  The special linear groups    2.5.4  The unitary groups    2.5.5  The orthogonal groups    2.5.6  The generalized orthogonal groups    2.5.7  The symplectic groups    2.5.8  The Heisenberg group    2.5.9  The Euclidean and Poincar6 groups  2.6  Properties of the Lie Algebra  2.7  The Exponential Mapping  2.8  Lie Algebras    2.8.1  Structure constants    2.8.2  Direct sums  2.9  The Complexification of a Real Lie Algebra  2.10 Exercises3  The Baker-Campbell-Hausdorff Formula  3.1  The Baker-Campbell-Hausdorff Formula for the Heisenberg Group  3.2  The General Baker-Campbell-Hausdorff Formula  3.3  The Derivative of the Exponential Mapping  3.4  Proof of the Baker-Campbell-Hausdorff Formula    3.5  The Series Form of the Baker-Campbell-Hausdorff Formula   3.6  Group Versus Lie Algebra Homomorphisms  3.7  Covering Groups  3.8  Subgroups and Subalgebras  3.9  Exercises4  Basic Representation Theory    4.1  Representations    4.2  Why Study Representations?  4.3  Examples of Representations    4.3.1  The standard representation    4.3.2  The trivial representation    4.3.3  The adjoint representation    4.3.4  Some representations of S(,1(2)      4.3.5  Two unitary representations of S0(3)    4.3.6  A unitary representation of the reals  ……Part II Semistmple TheoryReferencesIndex

用户评论 (总计22条)

•   Hall是表示论方向的大家幺
•   和好书籍，有收获的啊
•   建议购买！
•   这个是老师推荐用书，书很好。如果当当可以保证装订质量就好了
•   知识全面，帮助了解
•   这本书很好，很经典。比较适合初学者。
•   挺好的，推荐一下！
•   学习相关方向的数学系学生值得一看
•   挺好，很清晰
•   好书,但比较容易,初学合适
•   都17天了.书还没收到.怎么评价
•   这本书是从矩阵李代数讲起的，因此前提知识就是矩阵论的一些基本知识，很适合没有系统学习过流形的看起。推荐给想要了解李群和李代数但是没有太深背景的朋友阅读。
•   适合初学李群的人学习，有很多线性李群的例子。
•   价格合理，送货挺快，也有发票！赞一个！
•   只看了开头的一点儿，写得简单、明了，适合初学者。或许是非李群李代数专业的人作为了解性教材。GTM系列教材多是很经典的。
•   当时看到这个比较适合初学者就买了，结果发现是英文版的，还是慢慢啃吧
•   买了不少Lie group方面的书，很多都写得难懂！但这本书确实写的深入浅出！所以对于初学者，我强烈推荐这本书！
•   1、打开封面，接下来的一页破了个洞2、有的页面墨迹深，有的页面墨迹浅3、随便一翻，就发现第79页和第300页有不明记号总之，是一本盗版书
•   非常适合没有学过流形的同学， 比较初等的介绍矩阵群的性质
•   真不知道能不能看得懂？
•   The arefully choosed contents and exercises, suitble size to be hold in one's hands and the comfortable English writting, all these features make this book into a perfect one... 阅读更多
•   李群比较难学，这本书算是较简单的了。