Strade Completion of the Classification

III. Completion of the Classification
1. Auflage 2012
ISBN: 978-3-11-026301-5
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band Volume 3, 249 Seiten

Reihe: Simple Lie Algebras over Fields of Positive Characteristic

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 during the last 45 years 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 finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type.In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field.This is the last of three volumes. In this monograph the proof of the Classification Theorem presented in the first volume is concluded. It collects all the important results on the topic which can be found only in scattered scientific literature so far.

Zielgruppe

Researchers, Lecturers, Graduates, and Advanced in Algebra; Academic Libraries

Autoren/Hrsg.

Strade, Helmut

Fachgebiete

• Mathematik | Informatik Mathematik Algebra

Inhaltsverzeichnis

1;Introduction;9
2;16 Miscellaneous;11
2.1;16.1 Tori in some simple Lie algebras;11
2.2;16.2 Maximal tori in H(2; 1; .(1))[p];15
2.3;16.3 Representations of H(2; 1; .(t))(1);23
2.4;16.3.1 Central Extensions;25
2.5;16.3.2 Representations of dimension = p2;27
2.6;16.3.3 Splitting off the radical;39
2.7;16.4 Some properties of Melikian algebras;46
3;17 Sections;56
3.1;17.1 On trigonalizability;56
3.2;17.2 1-sections in simple Lie algebras of absolute toral rank 2;69
3.3;17.3 On the [p]-nilpotency of elements;81
3.4;17.4 2-sections;91
4;18 Solving the case when T is non-standard;103
4.1;18.1 2-sections revisited;103
4.2;18.2 Melikian pairs;115
4.3;18.3 Conclusion;119
5;19 Solving the case when all T-roots are solvable;129
5.1;19.1 2-sections revisited;129
5.2;19.2 The case when TR(L) = 3;131
5.3;19.3 Solvable sections;150
5.4;19.4 Conclusion;160
6;20 Attacking the general case;175
6.1;20.1 Optimal tori;176
6.2;20.2 Root spaces in 2-sections;186
6.3;20.3 The distinguished subalgebra Q(L,T);193
6.4;20.4 Pushing the classical case;199
6.5;20.5 The filtration defined by Q(L,T);203
6.6;20.6 Determining G(L,T);211
6.7;20.7 Completing the classification;229
6.8;20.8 Epilogue;245
7;Notation;247
8;Bibliography;249

Über Autor(innen)

Helmut Strade, University of Hamburg, Germany.