Introduces the concepts and methods of the Lie theory in a form accessible to the nonspecialist by keeping mathematical prerequisites to a minimum. Although the authors have concentrated on presenting results while omitting most of the proofs, they have compensated for these omissions by including many references to the original literature. Their treatment is directed toward the reader seeking a broad view of the subject rather than elaborate information about technical details. Illustrations of various points of the Lie theory itself are found throughout the book in material on applications.
In this reprint edition, the authors have resisted the temptation of including additional topics. Except for correcting a few minor misprints, the character of the book, especially its focus on classical representation theory and its computational aspects, has not been changed.
Author(s): J.G.F. Belinfante and B. Kolman
Series: Classics in Applied Mathematics
Edition: SIAM
Publisher: Society for Industrial Mathematics
Year: 1987
Language: English
Pages: 175
A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods......Page 2
ISBN 0-89871-243-2......Page 5
Preface to the Classic Edition......Page 6
Acknowledgments......Page 8
Contents......Page 10
Introduction......Page 14
1.1 THE GENERAL LINEAR GROUP......Page 16
1.2 ORTHOGONAL AND UNITARY GROUPS......Page 17
1.3 GROUPS IN GEOMETRY......Page 18
1.4 THE EXPONENTIAL MAPPING......Page 22
1.5 LIE AND ASSOCIATIVE ALGEBRAS......Page 23
1.6 LIE GROUPS......Page 24
1.7 LIE ALGEBRAS OF LIE GROUPS......Page 26
1.8 VECTOR FIELDS......Page 29
1.9 LIE THEORY OF ONE-PAR A METER GROUPS......Page 30
1.10 MATRIX LIE GROUPS......Page 32
1.11 POISSON BRACKETS......Page 35
1.12 QUANTUM SYMMETRIES......Page 37
1.13 HARMONIC OSCILLATORS......Page 41
1.14 LIE SUBGROUPS AND ANALYTIC HOMOMORPHISMS......Page 42
1.15 CONNECTED LIE GROUPS......Page 43
1.16 ABELIAN LIE GROUPS......Page 45
1.17 LOW-DIMENSIONAL LIE GROUPS......Page 46
1.18 THE COVERING GROUP OF THE ROTATION GROUP......Page 47
1.19 TENSOR PRODUCT OF VECTOR SPACES......Page 49
1.20 DIRECT SUMS OF VECTOR SPACES......Page 53
1.21 THE LATTICE OF IDEALS OF A LIE ALGEBRA......Page 54
1.22 THE LEVI DECOMPOSITION OF A LIE ALGEBRA......Page 55
1.23 SEMISIMPLE LIE ALGEBRAS......Page 56
1.24 THE BAKER-CAMPBELL-HAUSDORFF FORMULA......Page 58
2.1 LIE GROUP REPRESENTATIONS......Page 62
2.2 MODULES OVER LIE ALGEBRAS......Page 64
2.3 DIRECT SUM DECOMPOSITIONS OF LIE MODULES......Page 67
2.4 LIE MODULE TENSOR PRODUCT......Page 68
2.5 TENSOR AND EXTERIOR ALGEBRAS......Page 70
2.6 THE UNIVERSAL ENVELOPING ALGEBRA OF A LIE ALGEBRA......Page 74
2.7 NILPOTENT AND CARTAN SUBALGEBRAS......Page 76
2.8 WEIGHT SUBMODULES......Page 77
2.9 ROOTS OF SEMISIMPLE LIE ALGEBRAS......Page 78
2.10 THE FACTORIZATION METHOD AND SPECIAL FUNCTIONS......Page 81
2.11 THE CARTAN MATRIX......Page 84
2.12 THE WEYL GROUP......Page 85
2.13 DYNKIN DIAGRAMS......Page 87
2.14 IDENTIFICATION OF SIMPLE LIE ALGEBRAS......Page 89
2.15 CONSTRUCTION OF THE LIE ALGEBRA A2......Page 90
2.16 COMPLEXIFICATION AND REAL FORMS......Page 91
2.17 REAL FORMS OF THE LIE ALGEBRA A,......Page 94
2.18 ANGULAR MOMENTUM THEORY......Page 97
3.1 RAISING AND LOWERING SUBALGEBRAS......Page 102
3.2 DYNKIN INDICES......Page 104
3.3 IRREDUCIBLE REPRESENTATIONS OF A,......Page 106
3.4 THE CASIMIR SUBALGEBRA......Page 108
3.5 IRREDUCIBLE REPRESENTATIONS OF A2......Page 110
3.6 CHARACTERS......Page 112
3.7 COMPUTATION OF THE KILLING FORM......Page 114
3.8 DYNKIN'S ALGORITHM FOR THE WEIGHT SYSTEM......Page 117
3.9 FREUDENTHAL'S ALGORITHM......Page 120
3.10 THE WEYL CHARACTER FORMULA......Page 122
3.11 THE WEYL DIMENSION FORMULA......Page 125
3.12 CHARACTERS OF MODULES OVER THE ALGEBRA A2......Page 127
3.13 THE KOSTANT AND RACAH CHARACTER FORMULAS......Page 128
3.14 THE STEINBERG AND RACAH FORMULAS FOR CLEBSCH-GORDAN SERIES......Page 130
3.15 TENSOR ANALYSIS......Page 133
3.16 YOUNG TABLEAUX......Page 135
3.17 CONTRACTIONS......Page 139
3.18 SPINOR ANALYSIS AND CLIFFORD ALGEBRAS......Page 142
3.19 TENSOR OPERATORS......Page 148
3.20 CHARGE ALGEBRAS......Page 152
3.21 CLEBSCH-GORDAN COEFFICIENTS......Page 156
Bibliography......Page 160
Index......Page 170
