Malcev-Admissible Algebras

Description

Lie-admissible algebras emerged as generalizations of classical and quantum mechanics in 1967. Since then the subject has seen considerable growth both in mathemat- ics and theoretical physics and mechanics. This book presents a self-contained and detailed account of the mathematical theory' of Lie-admissible algebras which is based on the research of mathematicians and phy'sicists since 1977. It deals with the structure of a broader class of algebras, called Malcev-admissible alge- bras. which generalize Lie and Maleev algebras and which are also important algebraic models in the study of analy tic "-spaces. differential geometry', and invar- iant Lagrangian mechanics on homogeneous spaces. Since Lie-admissible algebras are a direct generalization of Lie algebras. the material in this volume will provide impetus for further development in modern theoretical phy'sics and many related areas. Many new simple alge- bras are presented, and open problems are suggested which will stimulate additional research in the area. This book is suitable supplementary reading for gradu- ate students as well as for research mathematicians and theoreticians. A comprehensive bibliography with mer 130 entries is included. CONTENTS PREFACE vii 1. FLEXIBLE MALCEV-ADMISSIBLE ALGEBRAS Introduction Basic results Cartan decompositions of A* Generalized Witt algebras Flexible Malcev-admissible nilalgebras 2. POWER-ASSOCIATIVE MALCEV-ADMISSIBLE ALGEBRAS 55 2.1. Introduction 56 2.2. Para-octonion and pseudo-octonion algebras 60 2.3. Power-associative products on matrices 72 2.4. Power-associative products on octonions 96 2.5. Power-associative products on simple Lie and Malcev algebras 109 2.6. The semisimple case 123 2.7. Power-associative products defined by linear forms 131 3. INVARIANT OPERATORS IN SIMPLE LIE ALGEBRAS AND FLEXIBLE MALCEV-ADMISSIBLE ALGEBRAS WITH A- SIMPLE 151 3.1. Introduction 150 3.2. Invariant operators 156 3.3. Modules for Malcev algebras 174 3.4. Adjoint operators in simple Lie algebras 183 3.5. Flexible Malcev-admissible algebras with A simple 192 4. MALCEV-ADMISSIBLE ALGEBRAS WITH THE SOLVABLE RADICAL OF A- NONZERO Derivation decompositions The case R is a direct summand of A Multiplication relations between irreducible summands Flexible Malcev-admissible algebras with abelian radical Quasi-classical Malcev algebras Wedderburn-type theory 5. MALCEV-ADMISSIBLE ALGEBRAS OF LOW DIr1ENSION 5.l. Basic results 5.2. Dimension 5 5.3. Dimension 6 5.4. Dimension 7 5.5. Dimension 8 5.6. Dimension 15 M'.(3) .::. Der A 5.7. Dimension 15 G 2 .::. Der A BIBLIOGRAPHY INDEX OF SYMBOLS INDEX OF TERMINOLOGY