INTRODUCTION xi We refer the interested reader to [St] for a comprehensive exposition of the classification. Recent work of Strade and the third author [PS1]-[PS5] has made sig- nificant progress on the problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic 7 and of characteristic 5, where the Generalized Kostrikin-Safareviˇ ˇ c Conjecture is known to fail because of the Melikyan algebras. (See Section 2.45.) The Classification Theorem an- nounced in [St, p. 7] (and also in [P2]), asserts “Every finite dimensional simple Lie algebra over an algebraically closed field of characteristic p 3 is of classical, Cartan or Melikyan type.” The Recognition Theorem plays a vital role in this extension of the classification theory to p = 5 and 7. In characteristics 2 and 3, many more simple Lie algebras which are nei- ther classical nor Cartan type are known (characteristic 3 examples can be found in [St, Sec. 4.4]). The papers [KO], [BKK], [BGK], and [GK] prove recognition theorems for graded Lie algebras of characteristic 3 under var- ious assumptions on the gradation spaces. One of the main challenges in characteristics 2 and 3 will be to remove such extra assumptions and deter- mine all finite-dimensional graded Lie algebras g satisfying conditions (b), (c), (d) above with the graded component g0 isomorphic to the Lie algebra of a reductive group. Once this is accomplished, one might be able to formu- late a plausible analogue of the Generalized Kostrikin-Safareviˇ ˇ c Conjecture for p = 2 and 3 and to begin the classification work in a systematic way. Our monograph consists of four chapters. In the first, we establish general properties of graded Lie algebras and use them to show that in a finite-dimensional graded Lie algebra satisfying conditions (a)-(d) of the Main Theorem, the representation of the commutator ideal g (1) 0 of the null component g0 on g−1 must be restricted. In Chapter 2, we gather useful in- formation about known graded Lie algebras, both classical and nonclassical. Chapter 3 deals with the case in which g−1 and g1 are dual g0-modules, the so-called contragredient case, which leads to the classical or Melikyan Lie algebras. Chapter 4 treats the noncontragredient case, and there the graded Lie algebras are shown to be of Cartan type. Acknowledgments Work on this monograph began under the sponsorship of National Science Foundation US-FSU Cooperative Program Grant #DMS-9115984, which funded the visits of Alexei Kostrikin, Michael Kuznetsov, and Alexan- der Premet to the University of Wisconsin-Madison. We are grateful for the Foundation’s support and for the enthusiasm that Alexei Kostrikin showed in urging us to undertake this project. We regret that he did not live to see its completion. We also wish to express our sincere thanks to Richard Block, Michael Kuznetsov, Hayk Melikyan, Yuri Razmyslov, George Selig- man, Serge Skryabin, Helmut Strade, and Robert Wilson for their interest and encouragement.

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