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
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