LIE ALGEBRAS

By T. RAERY

CONTENTS 1 INTRODUCTION DE?NITION OF LIE ALGEBRAS SOME EXAMPLES SUBALGEBRAS AND IDEALS HOMOMORPHISMS ALGEBRAS DERIVATIONS STRUCTURE CONSTANTS 2 IDEALS AND HOMOMORPHISMS CONSTRUCTIONS WITH IDEALS QUOTIENT ALGEBRAS CORRESPONDENCE BETWEEN IDEALS 3 LOW-DIMENSIONAL LIE ALGEBRAS DIMENSIONS 1 AND 2 DIMENSION 3 4 SOLVABLE LIE ALGEBRAS AND A ROUGH CLASSI?CATION SOLVABLE LIE ALGEBRAS NILPOTENT LIE ALGEBRAS A LOOK AHEAD 5 SUBALGEBRAS OF GL(V ) NILPOTENT MAPS WEIGHTS THE INVARIANCE LEMMA AN APPLICATION OF THE INVARIANCE LEMMA 6 ENGEL’S THEOREM AND LIE’S THEOREM ENGEL’S THEOREM PROOF OF ENGEL’S THEOREM ANOTHER POINT OF VIEW LIE’S THEOREM 7 SOME REPRESENTATION THEORY DE?NITIONS EXAMPLES OF REPRESENTATIONS MODULES FOR LIE ALGEBRAS SUBMODULES AND FACTOR MODULES IRREDUCIBLE AND INDECOMPOSABLE MODULES SCHUR’S LEMMA 8 REPRESENTATIONS OF SL(2, C) THE MODULES VD CLASSIFYING THE IRREDUCIBLE SL(2, C)-MODULES WEYL’S THEOREM 9 CARTAN’S CRITERIA JORDAN DECOMPOSITION TESTING FOR SOLVABILITY THE KILLING FORM TESTING FOR SEMISIMPLICITY DERIVATIONS OF SEMISIMPLE LIE ALGEBRAS ABSTRACT JORDAN DECOMPOSITION 10 THE ROOT SPACE DECOMPOSITION 101 PRELIMINARY RESULTS 102 CARTAN SUBALGEBRAS 103 DE?NITION OF THE ROOT SPACE DECOMPOSITION 104 SUBALGEBRAS ISOMORPHIC TO SL(2, C) 105 ROOT STRINGS AND EIGENVALUES 106 CARTAN SUBALGEBRAS AS INNER-PRODUCT SPACES 11 ROOT SYSTEMS 111 DE?NITION OF ROOT SYSTEMS 112 FIRST STEPS IN THE CLASSI?CATION 113 BASES FOR ROOT SYSTEMS 114 CARTAN MATRICES AND DYNKIN DIAGRAMS 12 THE CLASSICAL LIE ALGEBRAS 121 GENERAL STRATEGY 122 SL(?+ 1, C) 123 SO(2?+ 1, C) 124 SO(2?, C) 125 SP(2?, C) 126 KILLING FORMS OF THE CLASSICAL LIE ALGEBRAS 127 ROOT SYSTEMS AND ISOMORPHISMS 13 THE CLASSI?CATION OF ROOT SYSTEMS 131 CLASSI?CATION OF DYNKIN DIAGRAMS 132 CONSTRUCTIONS 14 SIMPLE LIE ALGEBRAS 141 SERRE’S THEOREM 142 ON THE PROOF OF SERRE’S THEOREM 143 CONCLUSION 15 FURTHER DIRECTIONS 151 THE IRREDUCIBLE REPRESENTATIONS OF A SEMISIMPLE LIE ALGEBRA 152 UNIVERSAL ENVELOPING ALGEBRAS 153 GROUPS OF LIE TYPE 154 KAC–MOODY LIE ALGEBRAS 155 THE RESTRICTED BURNSIDE PROBLEM 156 LIE ALGEBRAS OVER FIELDS OF PRIME CHARACTERISTIC 157 QUIVERS