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 neighbouring areas. This book is published in cooperation with Atlantic Association for Research in the Mathematical Sciences (AARMS).
Alberto Elduque, Universidad de Zaragoza, Zaragoze, SpainMikhail Kochetov, Memorial University of Newfoundland, St. John's, NL, Canada
Introduction Gradings on algebras Associative algebras Classical Lie algebras Composition algebras and type $G 2$ Jordan algebras and type $F 4$ Other simple Lie algebras in characteristic zero Lie algebras of Cartan type in prime characteristic Affine group schemes Irreducible root systems Bibliography Index of Notation Index