Introduction to Lie Algebras and Representation Theory (Google eBook)

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Springer Science & Business Media, 1972 - Mathematics - 169 pages
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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. 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
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XVI
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XVII
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XVIII
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XIX
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XX
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XXI
25
XXIII
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XXIV
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XXV
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XXVI
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XXVIII
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XXIX
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XXXI
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XXXII
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XXXIII
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XXXIV
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XXXV
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XXXVI
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XXXVII
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XXXVIII
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XL
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XLI
51
XLII
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XLIII
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XLV
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XLVI
57
XLVIII
63
L
65
LI
67
LIII
68
LIV
70
LV
73
LVI
74
LVII
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LVIII
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LX
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LXI
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LXII
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LXIV
82
LXVI
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LXVII
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LXVIII
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LXXVIII
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LXXIX
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LXXXII
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LXXXIII
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LXXXV
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LXXXVI
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LXXXVII
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LXXXVIII
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LXXXIX
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XC
115
XCI
117
XCII
118
XCIII
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XCIV
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XCV
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XCVI
124
XCVII
126
XCIX
128
C
130
CI
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CII
135
CIV
136
CV
138
CVI
140
CVII
143
CVIII
145
CX
146
CXI
148
CXII
149
CXIV
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CXV
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CXVI
153
CXVII
154
CXVIII
156
CXIX
157
CXXI
159
CXXII
161
CXXIII
162
CXXIV
163
CXXV
165
CXXVI
167
CXXVII
169
CXXVIII
172
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Page vii - 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...

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About the author (1972)

James E. Humphreys was born in Erie, Pennsylvania, and received his A.B. from Oberlin College, 1961, and his Ph.D. from Yale University, 1966. He has taught at the University of Oregon, Courant Institute (NYU), and the University of Massachusetts at Amherst (now retired). He visits IAS Princeton, Rutgers. He is the author of several graduate texts and monographs.

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