SEMI-SIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS

By WILLIAM WILSON

CONTENTS PART I GENERAL THEORY MATRIX LIE GROUPS DE?NITIONS EXAMPLES TOPOLOGICAL PROPERTIES HOMOMORPHISMS LIE GROUPS EXERCISES THE MATRIX EXPONENTIAL THE EXPONENTIAL OF A MATRIX COMPUTING THE EXPONENTIAL THE MATRIX LOGARITHM FURTHER PROPERTIES OF THE EXPONENTIAL THE POLAR DECOMPOSITION LIE ALGEBRAS DE?NITIONS AND FIRST EXAMPLES SIMPLE, SOLVABLE, AND NILPOTENT LIE ALGEBRAS THE LIE ALGEBRA OF A MATRIX LIE GROUP LIE GROUP AND LIE ALGEBRA HOMOMORPHISMS THE COMPLEXI?CATION OF A REAL LIE ALGEBRA THE EXPONENTIAL MAP CONSEQUENCES OF THEOREM 3 42 BASIC REPRESENTATION THEORY REPRESENTATIONS EXAMPLES OF REPRESENTATIONS NEW REPRESENTATIONS FROM OLD COMPLETE REDUCIBILITY SCHUR’S LEMMA REPRESENTATIONS OF SL 2I C/ GROUP VERSUS LIE ALGEBRA REPRESENTATIONS A NONMATRIX LIE GROUP THE BAKER–CAMPBELL–HAUSDORFF FORMULA AND ITS CONSEQUENCES THE “HARD” QUESTIONS AN ILLUSTRATIVE EXAMPLE THE BAKER–CAMPBELL–HAUSDORFF FORMULA THE DERIVATIVE OF THE EXPONENTIAL MAP PROOF OF THE BCH FORMULA THE SERIES FORM OF THE BCH FORMULA GROUP VERSUS LIE ALGEBRA HOMOMORPHISMS UNIVERSAL COVERS SUBGROUPS AND SUBALGEBRAS LIE’S THIRD THEOREM PART II SEMISIMPLE LIE ALGEBRAS THE REPRESENTATIONS OF SL 3I C/ PRELIMINARIES WEIGHTS AND ROOTS THE THEOREM OF THE HIGHEST WEIGHT PROOF OF THE THEOREM AN EXAMPLE: HIGHEST WEIGHT 1; 1/ THE WEYL GROUP WEIGHT DIAGRAMS FURTHER PROPERTIES OF THE REPRESENTATIONS SEMISIMPLE LIE ALGEBRAS SEMISIMPLE AND REDUCTIVE LIE ALGEBRAS CARTAN SUBALGEBRAS ROOTS AND ROOT SPACES ROOT SYSTEMS SIMPLE LIE ALGEBRAS THE ROOT SYSTEMS OF THE CLASSICAL LIE ALGEBRAS ABSTRACT ROOT SYSTEMS EXAMPLES IN RANK TWO DUALITY BASES AND WEYL CHAMBERS WEYL CHAMBERS AND THE WEYL GROUP DYNKIN DIAGRAMS INTEGRAL AND DOMINANT INTEGRAL ELEMENTS THE PARTIAL ORDERING EXAMPLES IN RANK THREE THE CLASSICAL ROOT SYSTEMS THE CLASSI?CATION REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS WEIGHTS OF REPRESENTATIONS INTRODUCTION TO VERMA MODULES UNIVERSAL ENVELOPING ALGEBRAS PROOF OF THE PBW THEOREM CONSTRUCTION OF VERMA MODULES IRREDUCIBLE QUOTIENT MODULES FINITE-DIMENSIONAL QUOTIENT MODULES THE STRUCTURE OF THE WEIGHTS THE CASIMIR ELEMENT THE WEYL CHARACTER FORMULA THE WEYL DIMENSION FORMULA THE KOSTANT MULTIPLICITY FORMULA THE CHARACTER FORMULA FOR VERMA MODULES PROOF OF THE CHARACTER FORMULA PART III COMPACT LIE GROUPS COMPACT LIE GROUPS AND MAXIMAL TORI TORI MAXIMAL TORI AND THE WEYL GROUP MAPPING DEGREES QUOTIENT MANIFOLDS PROOF OF THE TORUS THEOREM THE WEYL INTEGRAL FORMULA ROOTS AND THE STRUCTURE OF THE WEYL GROUP THE COMPACT GROUP APPROACH TO REPRESENTATION THEORY ANALYTICALLY INTEGRAL ELEMENTS ORTHONORMALITY AND COMPLETENESS FOR CHARACTERS THE ANALYTIC PROOF OF THE WEYL CHARACTER FORMULA CONSTRUCTING THE REPRESENTATIONS THE CASE IN WHICH I IS NOT ANALYTICALLY INTEGRAL FUNDAMENTAL GROUPS OF COMPACT LIE GROUPS THE FUNDAMENTAL GROUP FUNDAMENTAL GROUPS OF COMPACT CLASSICAL GROUPS FUNDAMENTAL GROUPS OF NONCOMPACT CLASSICAL GROUPS THE FUNDAMENTAL GROUPS OF K AND T REGULAR ELEMENTS THE STIEFEL DIAGRAM PROOFS OF THE MAIN THEOREMS THE CENTER OF K