 | Benkart G. The recognition theorem for graded lie algebras in prime characteristic / G.Benkart, T.Gregory, A.Premet. - Providence: American Mathematical Society, 2009. - xi, 145 p. - (Memoirs of the American Mathematical Society; Vol.197, N 920). - Bibliogr.: p. 143-145. - ISBN 978-0-8218-4226-3; ISSN 0065-9266
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Introduction ................................................... ix
Chapter 1. Graded Lie Algebras .................................. 1
1.1. Introduction ........................................... 1
1.2. The Weisfeiler radical ................................. 2
1.3. The minimal ideal  .................................... 4
1.4. The graded algebras  (V-t) and (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  0 on -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 0 ....................................... 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 [[-1, l],1] ≠ 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 К
series) ............................................... 54
2.12. The Recognition Theorem with stronger hypotheses ...... 56
2.13. ℓ as a 0-module for Lie algebras of Cartan type ... 57
2.14. Melikyan Lie algebras ................................. 66
Chapter 3. The Contragredient Case ............................. 69
3.1. Introduction .......................................... 69
3.2. Results on modules for three-dimensional Lie
algebras .............................................. 69
3.3. Primitive vectors in 1 and —1 ....................... 74
3.4. Subalgebras with a balanced grading ................... 77
3.5. Algebras with an unbalanced grading ................... 86
Chapter 4. The Noncontragredient Case .......................... 97
4.1. General assumptions and notation ...................... 97
4.2. Brackets of weight vectors in opposite gradation
spaces ................................................ 98
4.3. Determining 0 and its representation on -1 .......... 99
4.4. Additional assumptions ............................... 105
4.5. Computing weights of b--primitive vectors in 1 ...... 105
4.6. Determination of the local Lie algebra ............... 115
4.7. The irreducibility of 1 ............................. 125
4.8. Determining the negative part when 1 is
irreducible .......................................... 133
4.9. Determining the negative part when 1 is reducible ... 137
4.10. The case that 0 is abelian .......................... 141
4.11. Completion of the proof of the Main Theorem .......... 142
Bibliography .................................................. 143
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