## 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 | 2 |

V | 4 |

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

abelian adjoint representation admissible lattice arbitrary automorphism base Borel called Cartan matrix Casimir element char F Chevalley basis commute conjugate construction Corollary Coxeter graph cyclic module define denote derived diagonal diagram dimension direct sum dominant weights dual Dynkin diagram eigenvalues eigenvector endomorphisms Engel's Theorem Exercise exists fixed follows formula gl(K hence highest weight homomorphism implies induction integers invariant irreducible L-module isomorphism Jordan decomposition Killing form L-submodule Lemma linear Lie algebra maximal toral subalgebra maximal vector morphism nilpotent nondegenerate nonnegative nonzero orthogonal particular polynomial positive roots Proposition prove reader relative resp root system satisfying scalar multiplication semisimple Lie algebra shows simple roots solvable spanned standard basis subgroup submodule subspace tensor unique vector space verify weight spaces Weyl chamber Weyl group Weyl's Theorem write

### Popular passages

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...