Gradings on Simple Lie Algebras

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American Mathematical Soc., 2013 - Mathematics - 336 pages
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Gradings are ubiquitous in the theory of Lie algebras, from the root space decomposition of a complex semisimple Lie algebra relative to a Cartan subalgebra to the beautiful Dempwolff decomposition of $E_8$ as a direct sum of thirty-one Cartan subalgebras. This monograph is a self-contained exposition of the classification of gradings by arbitrary groups on classical simple Lie algebras over algebraically closed fields of characteristic not equal to 2 as well as on some nonclassical simple Lie algebras in positive characteristic. Other important algebras also enter the stage: matrix algebras, the octonions, and the Albert algebra. Most of the presented results are recent and have not yet appeared in book form. This work can be used as a textbook for graduate students or as a reference for researchers in Lie theory and neighboring areas.
  

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Contents

Introduction
1
Gradings on Algebras
9
Associative Algebras
27
Classical Lie Algebras
63
Composition Algebras and Type G2
123
Jordan Algebras and Type F4
163
Other Simple Lie Algebras in Characteristic Zero
207
Lie Algebras of Cartan Type in Prime Characteristic
271
Appendix A Affine Group Schemes
299
Appendix B Irreducible Root Systems
321
Index of Notation
331
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