李群，李代数和表示论

Part I General Theory

Matrix Lie Groups

1.1 Definition of a Matrix Lie Group

1.2 Examples of Matrix Lie Groups

1.3 Compactness

1.4 Connectedness

1.5 Simple Connectedness

1.6 Homomorpliisms and Isomorphisms

1.7 The Polar Decomposition for S[(n; R) and SL(n; C)

1.8 Lie Groups

1.9 Exercises

2 Lie Algebras and the Exponential Mapping

2.1 The Matrix Exponential

2.2 Computing the Exponential of a Matrix

2.3 The Matrix Logarithm

2.4 Further Properties of the Matrix Exponential

2.5 The Lie Algebra of a Matrix Lie Group

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 Exercises

3 The Baker-Campbell-Hausdorff Formula

4 Basic Representation Theory

Part II Semistmple Theory

References

Index

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 ones 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).