## Introduction to Lie Algebras and Representation TheoryThis book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry. |

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### Contents

II | 1 |

IV | 1 |

V | 1 |

VII | 6 |

VIII | 7 |

IX | 8 |

X | 10 |

XI | 11 |

LXIX | 89 |

LXX | 90 |

LXXI | 91 |

LXXII | 93 |

LXXIII | 94 |

LXXIV | 95 |

LXXV | 96 |

LXXVII | 98 |

XII | 12 |

XIII | 15 |

XV | 17 |

XVI | 19 |

XVII | 21 |

XVIII | 22 |

XIX | 23 |

XX | 24 |

XXI | 25 |

XXIII | 27 |

XXIV | 28 |

XXV | 29 |

XXVI | 31 |

XXVIII | 32 |

XXIX | 35 |

XXXI | 36 |

XXXII | 37 |

XXXIII | 38 |

XXXIV | 39 |

XXXV | 42 |

XXXVI | 43 |

XXXVII | 44 |

XXXVIII | 47 |

XL | 50 |

XLI | 51 |

XLII | 52 |

XLIII | 55 |

XLV | 56 |

XLVI | 57 |

XLVIII | 63 |

L | 65 |

LI | 67 |

LIII | 68 |

LIV | 70 |

LV | 73 |

LVI | 74 |

LVII | 76 |

LVIII | 78 |

LX | 79 |

LXI | 80 |

LXII | 81 |

LXIV | 82 |

LXVI | 83 |

LXVII | 84 |

LXVIII | 87 |

LXXVIII | 101 |

LXXIX | 102 |

LXXXII | 103 |

LXXXIII | 107 |

LXXXV | 108 |

LXXXVI | 109 |

LXXXVII | 112 |

LXXXVIII | 113 |

LXXXIX | 114 |

XC | 115 |

XCI | 117 |

XCII | 118 |

XCIII | 119 |

XCIV | 121 |

XCV | 123 |

XCVI | 124 |

XCVII | 126 |

XCIX | 128 |

C | 130 |

CI | 132 |

CII | 135 |

CIV | 136 |

CV | 138 |

CVI | 140 |

CVII | 143 |

CVIII | 145 |

CX | 146 |

CXI | 148 |

CXII | 149 |

CXIV | 151 |

CXV | 152 |

CXVI | 153 |

CXVII | 154 |

CXVIII | 156 |

CXIX | 157 |

CXXI | 159 |

CXXII | 161 |

CXXIII | 162 |

CXXIV | 163 |

165 | |

CXXVI | 167 |

169 | |

172 | |

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### Common terms and phrases

acts apply arbitrary associative associative algebra automorphism base basis called Chevalley clear closed combination commute conjugate consists construction contains Corollary corresponding course decomposition define definition denote derived determined dimension dominant elements endomorphisms equals example Exercise exists fact finite dimensional fixed follows forces formula function given hand hence highest weight homomorphism ideal implies includes induction integers Introduction invariant irreducible isomorphism L-module lattice Lemma Lie algebra lies linear matrix maximal vector module Moreover multiplication nilpotent nonzero Notes Notice obtain obvious occur particular polynomial positive possible Proof Proposition prove reader Recall relative representation result root system satisfying scalar semisimple sends shows simple solvable spanned standard standard basis subspace Theorem Theory toral subalgebra unique vector space verify weight Weyl write

### Popular passages

Page v - This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations.

Page 165 - On the Suzuki and Conway groups. In Representation theory of finite groups and related topics (Proc. Symp. Pure Math., vol. XXI, pp. 107-109. Providence, Rhode Island: American Mathematical Society.) Lindsey, II, JH 1971 6 A correlation between PSU4 (3), the Suzuki group, and the Conway group. Trans. Am. math. Soc. 157, 189-204. McKay, J. 1973 A setting for the Leech lattice. In Finite groups '72 (ed. T. Gagen et al...