Contents

Introduction ix

Chapter 1. Graded Lie Algebras 1

1.1. Introduction 1

1.2. The Weisfeiler radical 2

1.3. The minimal ideal I 4

1.4. The graded algebras B(V−t) and B(Vt) 5

1.5. The local subalgebra 8

1.6. General properties of graded Lie algebras 9

1.7. Restricted Lie algebras 15

1.8. The main theorem on restrictedness (Theorem 1.63) 17

1.9. Remarks on restrictedness 17

1.10. The action of g0 on g−j 18

1.11. The depth-one case of Theorem 1.63 20

1.12. Proof of Theorem 1.63 in the depth-one case 21

1.13. Quotients of g0 22

1.14. The proof of Theorem 1.63 when 2 ≤ q ≤ r 24

1.15. The proof of Theorem 1.63 when q r 25

1.16. General setup 25

1.17. The assumption [[g−1, g1], g1] = 0 in Theorem 1.63 30

Chapter 2. Simple Lie Algebras and Algebraic Groups 31

2.1. Introduction 31

2.2. General information about the classical Lie algebras 31

2.3. Representations of algebraic groups 38

2.4. Standard gradings of classical Lie algebras 41

2.5. The Lie algebras of Cartan type 42

2.6. The Jacobson-Witt algebras 43

2.7. Divided power algebras 44

2.8. Witt Lie algebras of Cartan type (the W series) 45

2.9. Special Lie algebras of Cartan type (the S series) 47

2.10. Hamiltonian Lie algebras of Cartan type (the H series) 50

2.11. Contact Lie algebras of Cartan type (the K series) 54

2.12. The Recognition Theorem with stronger hypotheses 56

2.13. g as a g0-module for Lie algebras g of Cartan type 57

2.14. Melikyan Lie algebras 66

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