Index

A 4 ,379 L,424 C2 ,369 L+,424 CN,373 SO(3, 1),483 D-functions, 329 Cl, C 2 , Cs, C 4 , C 6 ; D(Jo, C), 548 D 2 , D 3 , D 4 , D 6 ; Dj representation, 321 T; 0, 385 D 1/ 2 representation, 322 Ci == 8 2 : C 2h , C 4h , C6h : Djj', 538, 540 D 2h , D 4h , D 6h ; D 3 ,367 Th;Oh; GL(n,C),11 Dsv,Dsd; QR,14 Cs, 86,388 Qe,14 C 2h , C 4h , C6h ; 8L(2, C), 502, 504, 531, 538 D 2h , D 4h, D 6h; 8L(2, R), 506 Th,Oh,387 80(2,1), 505, 506 C2h,404 80(21),582 D 2h ,404 80(21 + 1),577,581 D 6h ,404 80(3), 274, 301, 369, 425, 505 80(3,1),426,502,505,522,548 80(3, C), 522 Abelian Group, 4 8U(I, 1),505 Abelian group, 171 8U(2), 136, 298, 505 Active point of view, 74 8U(1 + 1), 577, 580 Adjoint Representation, 574 84,382 Algebra 80(3), 304 8p(21), 578, 582 Alternating group An, 233 GL (n, c), 15 Antichronous transformation, SL(n,c),15 423 D 3 ,387 Associated Legendre functions, D 3d,387 343 D 3v , 387 Associativity, 4 L~, 425 Asssociated connection D 2 ,377 coefficients, 513 588 Linear Algebra and Group Theory

Balades, 462 Class, 20 Banach space, 546, 549 Class Multiplication, 163 Basis, 44 Class of conjugates, 19 Bijection, 2 Classes of the Symmetrie Bisymmetric, 264 Group,22 Bisymmetric , 257 Classification of OPLTs, 457 Blades of transformation, 466 Clebsch-Gordan (C - G), 325 rule, 56 Clebsch-Gordan coefficients Boost, 409 (Wigner coefficients), Boost operator, 409 350 Boost-like, 441, 458, 468 Clebsch-Gordan series, 349 Bra vector, 118 Clebsch-Gordan Theorem, 542, Brauer resolution, 289 543 Clifford-Dirac algebra, 36, 202 Canonieal basis, 313 Closure,4 Canonical coordinates of the Co-ordinate transformation, 73 first kind, 142 Cofactor, 12 Canonical coordinates of the Column vector, 40 second kind, 291 Commutative algebra, 259 Canonical Forms of Four• Commutative ring, 31 Vectors, 447 Commutator, 38 Canonical Forms of Planar Complex K, 15 OPLTs, 448 Complex Euclidean space, 87 Carrier space, 124 Complex Lie-Cartan Cartan-Weyl form, 577 Parameters of, 483,484 Casimir Operator, 573 Complex quaternions, 14 Casimir operators of Complex vector space, 35 SO(3, 1), 524 Components, 45 Cauchy sequence, 546 Composition law, 2 Cayley Hamilton Theorem, 81, Conjugate subgroup, 16, 370 459 Cosets, 17 Cayley's theorem, 28 Cramer's rule, 43 Cayley-Klein parameters, 503 Crystallattice, 384 Centre of a group, 16 Crystallographic point Centrum of the algebra, 193 groups, 367, 384 Character Formula, 321, 543 Cubic, 396 Character table, 168 Cyclie group, 15 Character Tables of the Point Cyclie permutation, 23 Groups, 395,399 Cyclie subgroup, 18, 371 Characteristic equation, 79 Characteristic polynomial, 79 Dihedral group D 3 , 8,390 Classification oE Lie Groups 589

Dihedral group D 4 , 392 First rank spinors, 508 Dihedral group D6 , 393 Four-group of Klein, 27 Dihedral group D n , 374 Four-veetors, 413 Dihedral group D2 , 29 Frobenius algebra, 37 Dihedral group D 3 , 8 Frobenius Criterion, 160, 174 Dimension of a Vector spaee, 42 Frobenius Criterion for Dirae, 118 Irredueibility, 159 Dirae a-matriees, 139 Fundamentalsequenee, 546 Dirae algebra, 206 Fundamental Theorem, 53, 184 Dirae Algebras C2 and C4 , 133 Dirae groups, 175 Gelfand-Naimark Direct product, 24, 98 Basis, 545, 548 Direet product of algebras, 225 General Linear Group GL(n, c), Direet produet of two groups, 24 13 Direet sum, 51, 127 General Lorentz Group, 143 Distributive law, 31 General Theory of Division ring, 34 Relativity, 445 Dotted spin spaee, 508 Generalised Schur Lemma, 147 Duffin-Kemmer-Petiau Generating idempotent, 184 relations, 208 Generating system, 53 Dynkin Diagrams, 578 Geometrical realisation, 381 Dynkin diagrams, 569 Geometrieal Representation of 80(3, 1), 500 Eigenvalue, 78, 279 Great orthogonality theorem, Eigenveetor, 78 157 Eleetromagnetie Group, 3 field tensor, 457, 521 Group axioms, 4 Equivalent left ideals, 194 Group of symmetries, 7 Equivalent poles, 372 Group-Ring, 37 Equivalent Representations, 124 Euelidean spaee Es, 500 Hamermesh, 395 Euler angles, 285 Hamiltonian eonjugate, 14, 490 Euler Resolution, 283, 468 Hermitian adjoint, 91 Euler-Brauer resolution, 291, Hermitian form, 91 548 , 91, 111 Even permutation, 23 Hermitian operator, 114 Exeeptional, 458 Hexagonal, 396 Hilbert spaee, 546 Factor group, 18, 26, 123 Homogeneous Linear Equation, Field,34 60 Finite group, 151 Homomorphism, 25 590 Linear Algebra and Group Theory

Hyper complex system, 36 Left ideal, 180, 191 Left translations, 301 Icosahedral group, 375 Lie algebra (80(3,1),523 Idempotent, 184 Lie group, 139, 569 Identity, 4 Lie-algebra, 37 Identity operator, 119 Lie-Cartan integrability Identity Representation, 124 conditions, 141 ILT,459 Lie-Cartan parameters, 503 Index diagrams, 256 Lie-ring, 37 Inequivalent representation, 124 Linear associative algebra, Infinite group, 4 36, 38, 180 Infinitesimal Operators, 535 Linear Dependence, 41 Infinitesimal Linear endomorphism, 64 transformation, 140 Linear fractional Injection, 2 transformation, 300 Integral algebraic number, 167 Linear huH, 53, 254 Intersection, 1, 51 Linear transformation, 64 Invariant Integral, 301 Linear Vector Space, 35, 39 Invariant subgroup, 16 , 41 Inverse, 4 Lorentz frames, 522 Inverse image, 2 Lorentz group, 407, 420, 500 Involution, 278, 443 Lorentz Matrix, 109 Irreducible tensor, 337 Lorentz transformation, 109, Irreps, 127 407, 420 Isomorphie, 54 Isomorphism, 25 Main Theorem, 184 Mashke's Theorem, 154 Jacobi's Theorem, 473 , 105 Matrix of passage, 49 Kemmer Algbera K 3 , 209 Matrix operator, 70 Kemmer Algebra K 4 , 209, 218 Matrix product rule, 55 Kemmer equation, 208 Matrix ring, 33 Kernel,75 Minimal condition, 182 Ket vector, 118 Minimal equation, 47, 86 Klein's Vierer gruppe, 27 Minimalleft ideal, 181 Kronecker product, 98, 100,542 Minimal polynomial, 47 Kronecker Product Representa- Minimal right ideal, 181 tion, 132 Minkowski coordinates, 110, 444 Minkowski Space, 413 Lagrange's theorem, 17 Minkowski vectors, 413 Left coset, 17 Minkowskian coordinates, 474 Classification oE Lie Groups 591

Minkowskian eigenvectors, 459 Orhtochronous Proper Lorentz Minkowskian quaternion, 491 Group, 146, 425 Minkowskian space-time, 490 Orthochronous ,57 transformation, 423 Minquat, 490 , 67 Modulus,87 Orthogonality, 90 Monoclinie, 396 Orthogonality relations, 155 Multiplicative group, 176 Orthonormal basis, 93 Murnaghan, 403 Orthonormal set, 90 Orthorhombic, 396 Nilpotent, 181 Non-commutative ring, 31 Passive point of view, 75 Non-null Lorentz , 135, 139 transformation, 458 Permutation, 9 Non-Null OPLT, 451 Pierce Decomposition, 185 Non-null OPLT, 452, 468 Planar OPLT, 487 Norm, 14 Planar transformation, 434 Normal matrices, 92 Pointwise invariant plane, 440 Normal subgroup, 16, 26 Pole of order n, 368 Normal transformation, 93 Primitive, 189 Normalised to unity, 87 Primitive idempotent, 199 Normaliser of a complex, 19 Primitive translations, 45 Normed linear space, 546 Principal minor, 117 Null, 417, 458 Product polynomial, 32 Null set, 1 Proper Lorentz group, 424 Null space, 75 Proper orthogonal matrix, 278, Nullity, 76 474 Proper subgroup, 15 Octahedral group, 375 Pseudo-unitary, 552 Odd permutation, 23 Pseudo-unitary representations, One-column tableau, 271 562 One-row tableau, 271 Pure quaternion, 292 One-to-one, 76 Operator algebra, 71 Quaternion algebra, 135 Operator Homomorphism, 182 Quaternion groups Qc, 13 Operator irreducible, 546 Quaternion units, 13 Operator-irreducibility, 148 Quaternions, 291 Operator-isomorphism, 183 Quotient group, 18 OPLT, 427, 447, 451, 459 Order of a group, 17 Range of the operator Ci, 75 Order of the element, 15 Rank,53 592 Linear Algebra and Group Theory

Rank of matrix, 57 Schur eanonieal form, 94, 167 Rational parametrisation Schur lemma, 129, 530 of the rotation group, Sehur's Theorem, 94 280 Schwarz inequality, 89 Real Euelidean spaee, 87 Serew-like OPLT, 462, 487 Real linear veetor spaee, 35 Self-adjoint, 91 Real number field, 108 Semi-group, 3 Real orthogonal matrix, 92 Semi-simple, 182, 569 Real quaternion, 14 Semisimple Lie algebra, 571 Real skew-symmetrie matrix, 92 Simple, 569 Real symmetrie matrix, 92 Simple algebra, 181, 571 Reality classifieation, 562 Simple roots, 87 Reciproeity Theorem, 266 Skew symmetrie matrix, 51 Reflection in a plane, 69 Skew-field, 35 Regular Representation, Spaee-like, 417 134, 179,575 Special unitary group of Relativity theory, 412 dimension, 136 Representation of a linear group Spherieal eomponents, 320 G,146 Spherieal tensor, 337 Representation of the Symmet- Spin spaee, 507 rie group, 231 Spinor Algebra, 508 Rhombohedral, 396 Spinors, 507 Right eoset, 17 Spur, 73 Right ideal, 180, 191 Standard Tableaux, 248 Right identity, 3 Stereographie Right inverse, 3 Projeetion, 298, 395 Ring, 31 Strueture eonstants, 141 Ring of eomplex quaternions, 33 Subfield, 35 Ring of integers, 32 Subgroup, 6, 15 Ring of polynomials, 32, 33 Subspace, 50 Ring of real quaternions, 34 Subspaee-irredueible, 546 , 66, 273, 276 Symmetrie group Sn, 9, 11 Rotation-like, 458, 468 Symmetry classes, 255 Rotation-reflection matrix, 70 Symmetry operation, 7 Row veetor, 40 Sympleetie group*, 333 Running coordinates, 284 Sympleetie representation, 552 Synge, 458 Scalars, 40 Synge-serew form, 452 Sehmidt Orthogonalisation Proeedure, 90 Tangent vectors, 291 Sehoenflies notation, 403 Tetragonal, 396 Classincation oE Lie Groups 593

Tetrahedral group, 375 Zero divisors, 31 The Euler resolution of an Zero polynomial, 32 OPLT, 476 The Euler-Brauer resolution of an OPLT, 479 The Regular Representation, 161 Theorem of Cartan, 573 Theorem of Young, 237 Theory of relativity, 407 Thomas Precession, 498 Time-like, 417 Transpose conjugate, 88 Transposition, 23 Triangular form, 94 Triclinic, 396 Two-sided ideal, 181 Two-sided ideals, 191

Unimodular unitary group of dimension, 136 Unit complex quaternion, 490, 500 Unit real quaternion, 494, 500 Unitaryoperator, 115 Unitary representation, 151 Unitary space, 87 Unitary transformation, 93

Van der Waerden connection C0- efficients, 513 Vector-Coupling coefficients, 350 Vectors,40

Wedderburn's Theorem, 197 Wigner-Eckart Theorem, 354

Young frame, 237, 253 Young Symmetriser, 247 Young Tableaux, 234