Simple Lie Algebras over Fields of Positive Characteristic Volume 2 Classifying the Absolute Toral Rank Two Case 2nd Edition by Helmut Strade 9783110517606 3110517604
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Product details:
ISBN 10: 3110517604
ISBN 13: 9783110517606
Author: Helmut Strade
The problem of classifying the finite dimensional simple Lie algebras over fields of characteristic p > 0 is a long standing one. Work on this question has been directed by the Kostrikin Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every simple finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. This is the second part of a three-volume book about the classification of the simple Lie algebras over algebraically closed fields of characteristic > 3. The first volume contains the methods, examples and a first classification result. This second volume presents insight in the structure of tori of Hamiltonian and Melikian algebras. Based on sandwich element methods due to A. I. Kostrikin and A. A. Premet and the investigations of filtered and graded Lie algebras, a complete proof for the classification of absolute toral rank 2 simple Lie algebras over algebraically closed fields of characteristic > 3 is given. Contents Tori in Hamiltonian and Melikian algebras 1-sections Sandwich elements and rigid tori Towards graded algebras The toral rank 2 case
Table of contents:
10 Tori in Hamiltonian and Melikian algebras
10.1 Determining absolute toral ranks of Hamiltonian algebras
10.2 More on H(2; (1,2))(2)[p]
10.3 2-dimensional tori in H(2; 1; Φ(τ))(1)
10.4 Semisimple elements in H(2; 1; Φ(1))[p]
10.5 Melikian algebras
10.6 Semisimple Lie algebras of absolute toral rank 1 and 2
10.7 Weights
11 1-sections
11.1 Lie algebras of absolute toral rank 1
11.2 1-sections
11.3 Representations of dimension < p2
11.4 More on H(2; 1)(2)
11.5 Low dimensional representations of H(2; 1)(2)
12 Sandwich elements and rigid tori
12.1 Deriving identities
12.2 Sandwich elements
12.3 Rigid roots
12.4 Rigid tori
12.5 Trigonalizability
13 Towards graded algebras
13.1 The pentagon
13.2 An upper bound
13.3 Filtrations
13.4 More on Hamiltonian roots
13.5 Switching tori
13.6 Good triples
13.7 On the existence of good tori and good triples
14 The toral rank 2 case
14.1 No root is exceptional
14.2 S is not of Cartan type
14.3 Graded counterexamples
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Tags: Helmut Strade, Simple, Algebras, Positive