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Weyl H. — The Classical Groups: Their Invariants and Representations, Vol. 1
Weyl H. — The Classical Groups: Their Invariants and Representations, Vol. 1

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Название: The Classical Groups: Their Invariants and Representations, Vol. 1

Автор: Weyl H.


In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. Analysis and topology are used wherever appropriate. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics.

Hermann Weyl was among the greatest mathematicians of the twentieth century. He made fundamental contributions to most branches of mathematics, but he is best remembered as one of the major developers of group theory, a powerful formal method for analyzing abstract and physical systems in which symmetry is present. In The Classical Groups, his most important book, Weyl provided a detailed introduction to the development of group theory, and he did it in a way that motivated and entertained his readers. Departing from most theoretical mathematics books of the time, he introduced historical events and people as well as theorems and proofs. One learned not only about the theory of invariants but also when and where they were originated, and by whom. He once said of his writing, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful."

Weyl believed in the overall unity of mathematics and that it should be integrated into other fields. He had serious interest in modern physics, especially quantum mechanics, a field to which The Classical Groups has proved important, as it has to quantumchemistry and other fields. Among the five books Weyl published with Princeton, Algebraic Theory of Numbers inaugurated the Annals of Mathematics Studies book series, a crucial and enduring foundation of Princeton's mathematics list and the most distinguished book series in mathematics.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1946

Количество страниц: 320

Добавлена в каталог: 01.06.2008

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Предметный указатель
Absolute (the)      254
Absolute (the) and relative invariant      25 263
Absolute (the) coordinate system      8
Absolute (the) irreducibility      92
Abstract algebra      79
Abstract group      14
Adaptation of coordinate system      9
Addition of representations      19
Adjoint realization and representation      190
Adjunction of roots      289
Adjunction of the absolute      255
Affine invariants      47 52 254
Algebra product      284 286
Algebra, abstract, and of matrices      79
Algebra, abstract, and of matrices (orthogonal)      141
Algebra, abstract, and of matrices (symplectic)      174 145
Algebra, abstract, and of matrices, $ \Re_{f}$,      136
Algebra, abstract, and of matrices, $\omega^{n}_{f}$      148
Almost periodic      193
Alternating group      34
Alternation      120
Angles of unitary transformations      180 217 223
Automorphism, in general      15
Automorphism, in general of algebras      281 ff
Averaging over a group      185
Axis of a rotation      58
Basis of a linear set      6
Basis of a linear set, ideal      3
Basis of a linear set, integrity      30
Betti numbers      277
Betti numbers of compact Lie groups and the classical groups      279
Binary quadratic      246
Bisymmetric      98
Brauer product      287
Capelli's formal congruence      73
Capelli's identities      42
Cartesian coordinates      12
Cauchy's lemma      202
Cayley parametrization      56 169 171 177
Centrum      92 280
Character of representation      19
Character of representation for alternation and symmetrization alone      185
Character of representation of representation induced by regular one      105
Character of representation of the linear group      203 204
Character of representation of the orthogonal group      228 268
Character of representation, of spin representation      273
Character of representation, symmetric group      213
Character of representation, symplectic group      218 219
Character of representation, unitary group      200 201
Characteristic of a field      2
Characteristic of a field, of a representation of the symmetric group      215
Characteristic, polynomial      8
Class function      105 198
Classic invariants and covariants      239
Cogredient and contragradient      10 21
Collineation      112
Commutative set of transformations      180
Commutator      81
Commutator, algebra      81
Compactness      178
Complete matrix algebra      86
Completeness for representations      116 189
Complex symbol      51
Congruence modulo ideal      2
Congruence modulo ideal, of geometric figures      22
Connectivity of the unitary group      268
Connectivity of the unitary group, of the other classical groups      270
Coordinate system      6
Coordinate system, absolute      8
Covariant and contravariant tensors      132
Covariant and contravariant vectors      10
Covariant in the general sense      25
Covariant in the general sense, in the classical sense      239
Decomposition of a matrix      9
Decomposition of a matrix, of sets of matrices, in particular representations      19
Decomposition of tensor space under the linear group      129 130 206—208
Decomposition of tensor space under the linear group, orthogonal group      150 157 158 164 229
Decomposition of tensor space under the linear group, symplectic group      175 221 222
Decomposition of the representations of the classical groups and their Kronecker products      232
Degeneracy of first and second kind      85
Degree of a polynomial      3
Degree of a polynomial, of matrix and matric set      8
Derivative of a polynomial      4
Derivative of a polynomial, of differentials      277
Diagram      120
Differential      276
Differential equations for invariants      262
Differential, exact      277
Differential, total      5
Dimensionality      7
Direct sum      93
Division algebra      80 87
Double-valued representations of the orthogonal group      268
Dual spaces      10
Elementary sum, for the unitary group      199
Elementary sum, for the unitary group, orthogonal group      224 227
Elementary sum, for the unitary group, symplectic group      218
Enveloping algebra      79
Enveloping algebra, for the linear group      130
Enveloping algebra, of a fully reducible matric set      95
Enveloping algebra, the orthogonal group      141 143 145
Enveloping algebra, the symplectic group      174
Equivalent subspaces      103
Euclidean field      273
Even and odd invariants      53
Exact differential      277
Exceptional matrix      58
Expansions in the theory of invariants      135
Extension of field      284 289
Extension theorem      47
Faithful      14 80
Field      1
Field, Euclidean      273
Field, Pythagorean      13
Field, real      13
First Main Theorem      30 see
FORM      5
Form basis      299 (chapter X)
Formal (orthogonal) invariant      63
Formal (orthogonal) invariant, (symplectic)      172
Frame      17
Fully reducible matric algebra and matric set      94 95
Generating idempotent      88 101
Gram's theorem      240
Gram's theorem, generalized      242
Ground field      2
Group      14
Group germ      258
Group of plane rotations      192
Group of step transformations      48
Group ring      97
Group ring, its full reduction      101
Group ring, modified      114
Group, all other special groups      see under their characteristic adjectives
Group, its invariants      257
Group, its vector invariants      49 52
Hermitian form      170
hessian      240
Homology      277
Hubert's theorem on ideal bases      36 251
Ideal      2
Ideal of a group of linear transformations      299 (chapter X)
Ideal, basis of an ideal      3
Ideal, orthogonal      143 147
Ideal, polynomial      251
Ideal, prime      3
Ideal, principal      3
Ideal, symplectic      174
Idempotent      85
Idempotent, generating      101
Identical representation      18
Improper orthogonal transformation      11
Induced transformation      10 96
Infinitesimal elements of a group      68 260
Infinitesimal rotations      67
Infinitesimal vector invariants for the orthogonal group      293 (chapter X)
Infinitesimal vector invariants for the orthogonal group, for the symplectic group      294 (chapter X)
Integrity basis      30
Invariant differential      277
Invariant subspace      10 18
Invariant, absolute and relative      25 263
Invariant, absolute and relative, (symplectic)      172
Invariant, absolute and relative, formal (orthogonal)      63
Invariant, absolute and relative, infinitesimal      68 262
Invariant, absolute and relative, special and general      23 24
Invariants of the adjoint group      237
Invariants of the adjoint group, binary cubic      249
Invariants of the adjoint group, finite groups      275
Invariants of the adjoint group, for all classical groups and arbitrary Lie representations      275
Invariants of the adjoint group, for compact Lie groups      274
Invariants of the adjoint group, in affine space      257
Invariants of the adjoint group, of the binary quadratic      246
Invariants of the adjoint group, of the linear group      254
Invariants of the adjoint group, of the orthogonal and symplectic groups      257—258 275
Invariants of the adjoint group, quadratic      248
Inverse algebra      90
Irreducible      19
Jacobian      240
Klein's Erlanger program      14 28
Kronecker product of algebras      286—288
Kronecker product of matrices and representations      20
Kronecker product of matrices and representations, its decomposition in the case of the classical groups      232
Left and right invariant      103
Lie algebra      260
Lie group      187
Lie representation      264
Linear closure      79
Linear form      8
Linear group      13
Linear group,      138
Linear group, its invariants      254 275
Linear group, representations      129—130
Linear group, representations,      266—267
Linear group, representations, character      203
Linear group, representations, enveloping algebra      130
Linear group, second main theorem      70
Linear group, their enumeration      207
Linear group, vector invariants      45—47
Linear mapping      7
Linear set      6
Lorentz group      66
Matric algebra      79
Matrix      7
Metric ground form      65
Multiplication of algebras      286
Multiplication of representations      20 232
Multiplier      25
Multiplier, infinitesimal      263
Norm of a matrix      191
Normal idempotents      102
Null algebra      85
Null divisor      2
Octahedral group      218
Odd invariants      53
Order of a linear set      7
Orthogonal group, proper and improper,      13
Orthogonal group, proper and improper, connectivity      269
Orthogonal group, proper and improper, enveloping algebra      141 143 145
Orthogonal group, proper and improper, invariants      257 258 275
Orthogonal group, proper and improper, vector invariants      52
Orthogonal group, proper and improper, vector invariants, formal      63
Orthogonal group, proper and improper, vector invariants, infinitesimal      68
Orthogonal group, proper and improper, vector invariants, second main theorem      75 77
Orthogonal group, proper and improper, vector invariants, their enumeration      229
Orthogonal group, proper and improper, volume element      224 226
Orthogonal group, proper and improper, with arbitrary ground form      65
Orthogonal group, proper and improper,, Cayley's parametrization      56
Orthogonal group, representations      157—158 164 228 268
Orthogonal group, representations, characters      225 227 228 268
Orthogonal ideal      143 147
Orthogonal ideal (full and proper)      299 (chapter X)
Orthogonal transformation      11
Orthogonal transformation, transformation to principal axes      223
Orthogonality of representations      115
Parallelism between group ring and commutator algebra      107 110
Parametrization of the orthogonal group      57
Parametrization of the orthogonal group, of the symplectic group      169
Parseval equation      192
Partition      120
Pascal's theorem      251
Peiree decomposition      85
Permutation      119
Pfaffian      166
Poincare polynomial      233
Poincare polynomial, of the linear group      233
Poincare polynomial, of the other classical groups      238
Polarization      5
Polynomial      3
Polynomial ideal      251
Prime ideal      3
Primitive idempotent      102
Primitive quantity      19
Principal axes      179 217 223
Principal ideal      3
Projection      101
Projective geometry      112—113
Projectivity      113
Proper orthogonal transformation      11
Pythagorean adjunction      60
Pythagorean field      13
Quantic      132
quantity      17 18
quantum mechanics      99
Quasi-unimodular      190
Quotient field      2
Rank of a tensor      21
Real field      13
Realization of a group      14
Reduction      18
Regular realization of a group      14
Regular representation of an algebra      80
Relation      36
Relative invariant      25 263
Relativity problem      16
Representation of an algebra      80
Representation of an algebra, (regular)      80
Representation of an algebra, of a group      14
Representation of an algebra, of a group, identical      18
Representations of the linear group      129—130 266—267
Representations of the linear group, of the full orthogonal group      157—158 228 268
Representations of the linear group, of the proper orthogonal group      164 268
Representations of the linear group, of the symmetric group      124—127
Representations of the linear group, of the symplectic group      174 267
Restituent and restitution      50
Right and left invariant      103
Ring      2
Scalarproduct      12
Schur's lemma      81 83
Second Main Theorem      36 254
Second main theorem, for the linear group (vector invariants)      70
Second main theorem, the orthogonal group      75 77
Second main theorem, the symplectic group      168
Semi-group      79
Semi-invariant      49
Semi-linear substitution      113
Signature      131 132
Similarity mapping      103
Similitude      15
Simple algebra      85
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