李代数是一类重要的非结合代数，随着时间的推移，李代数在数学以及古典力学和量子力学中的地位不断上升，其理论也在不断完善和发展，很多理论与方法已经渗透到了数学和理论物理的许多领域。本书采用大胆而新颖的方法对李代数及其表示进行了论述。本书共分八章，从对李代数概念的介绍入手，阐述了李代数及其表示的相关性质及理论，重点介绍了李代数在表示论中取得的一个重要成果——一般线性李代数不可约模的最高权分类。本书适合大学师生、研究生及数学爱好者参考阅读。

Preface
Notationaf conventions
1 Motivation：representations of Lie groups
1.1 Homomorphisms of general linear groups
1.2 Multilinear algebra
1.3 Linearization of the problem
1.4 Lie’s theorem
2 Definition of a Lie algebra
2.1 Definition and first examples
2.2 Classification and isomorphisms
2.3 Exercises
3 Basic structure of a Lie algebra
3.1 Lie subalgebras
3.2 Ideals
3.3 Quotients and simple Lie algebras
3.4 Exercises
4 Modules over a Lie algebra
4.1 Definition of a module
4.2 Isomorphism of modules
4.3 Submodules and irreducible modules
4.4 Complete reducibility
4.5 Exercises
5 The theory of sl2-modules
5.1 Classification of irreducibles
5.2 Complete reducibility
5.3 Exercises
6 General theory of modules
6.1 Duals and tensor products
6.2 Hom—spaces and bilinear forms
6.3 Schur’s lemma and the Killing form
6.4 Casimir operators
6.5 Exercises
7 Integral gln—modules
7.1 Integral weights
7.2 Highest—weight modules
7.3 Irreducibility of highest—weight modules
7.4 Tensor-product construction of irreducibles
7.5 Complete reducibility
7.6 Exercises
8 Guide to further reading
8.1 Classification of simple Lie algebras
8.2 Representations of simple Lie algebras
8.3 Characters and bases of representations
Appendix SolutionS to the exercises
Solutions for Chapter 2 exercises
Solutions for Chapter 3 exercises
Solutions for Chapter 4 exercises
Solutions for Chapter 5 exercises
Solutions for Chapter 6 exercises
Solutions for Chapter 7 exercises
References
Index
编辑手记