Representation of Lie groups and special functions: 3 Classical and quantum groups and special functions
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer Academic Publications
1992
|
| Schriftenreihe: | Mathematics and its Applications. Soviet Series
75 |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 629 Seiten |
| ISBN: | 079231493X |
Internformat
MARC
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| 035 | |a (OCoLC)312016259 | ||
| 035 | |a (DE-599)BVBBV005847546 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-384 |a DE-91G |a DE-12 |a DE-739 |a DE-824 |a DE-29T |a DE-188 | ||
| 100 | 1 | |a Vilenkin, Naum Ja. |d 1920-1991 |e Verfasser |0 (DE-588)127328122 |4 aut | |
| 245 | 1 | 0 | |a Representation of Lie groups and special functions |n 3 |p Classical and quantum groups and special functions |c by N. Ja. Vilenkin and A. U. Klimyk |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Academic Publications |c 1992 | |
| 300 | |a XIX, 629 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its Applications. Soviet Series |v 75 | |
| 490 | 0 | |a Mathematics and its Applications / Soviet Series |v ... | |
| 700 | 1 | |a Klimyk, Anatolij U. |d 1939-2008 |e Verfasser |0 (DE-588)115774580 |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV005437238 |g 3 |
| 830 | 0 | |a Mathematics and its Applications. Soviet Series |v 75 |w (DE-604)BV004708148 |9 75 | |
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| 940 | 1 | |n oe | |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-003661615 | ||
Datensatz im Suchindex
| _version_ | 1804120037661868032 |
|---|---|
| adam_text | Table of Contents
Series Editor s Preface
List of Special Symbols xvii
Chapter 14:
Quantum Groups, g Orthogonal Polynomials and
Basic Hypergeometric Functions 1
14.1. g Analysis and Basic Hypergeometric Functions 1
14.1.1. g Factorials 1
14.1.2. The tnnction i(fio(a;q,x) 3
14.1.3. Expressions for (a;q)n, (a;^) 1 and
their corollaries 5
14.1.4. q Analog of the binomial formula 8
14.1.5. ^ Differentiation and g integration 9
14.1.6. g Analogs of the exponential and of
the trigonometrical functions 12
14.1.7. g Analogs of the gamma and beta functions 14
14.1.8. Properties of the basic hypergeometric
function 2 Pi 17
14.2. Hopf Algebras, Their Representations and
Corepresentations 20
14.2.1. Introduction 20
14.2.2. Algebra of functions on a group 21
14.2.3. Definition of a Hopf algebra 23
14.2.4. Coordinate functions 25
14.2.5. Representations and corepresentations of Hopf algebras 26
14.3. Representations of the Quantum Algebra t/^sfe)
and Its Clebsch Gordan Coefficients 29
14.3.1. The quantum algebra f7j(sb) and
its real forms 29
14.3.2. Finite dimensional representations of Z7?(sl2) 32
14.3.3. The tensor product of representations 34
14.3.4. Calculation of CGC s 36
14.3.5. Expressions for CGC s in terms of the function 3$2 • ¦ 38
14.3.6. Special cases of CGC s 39
14.3.7. Symmetries of CGC s 40
14.3.8. Generating functions for CGC s 41
14.3.9. The difference equation for CGc s 45
14.3.10. Recurrence relations for CGC s 45
14.4. Matrix Elements of Representations of t/^sfe) 47
vii
viii Table of Contents
14.4.1. Introduction 47
14.4.2. Relations for tt^ 48
14.4.3. Calculation of matrix elements 49
14.4.4. Expressions in terms of g Jacobi polynomials 51
14.5. Racah Coefficients of the Algebra Uq(si2) 52
14.5.1. Properties of Racah coefficients 52
14.5.2. Calculation of RC s 53
14.5.3. Special values of RC s 56
14.5.4. The Biedenharn Elliott identity 57
14.5.5. The addition theorem for RC s 58
14.5.6. Generalization of the Biedenharn Elliott identity ... 59
14.5.7. CGC s as a limit of RC s 60
14.5.8. Other asymptotic formulas for RC s 61
14.5.9. Recurrence relations and the second order
difference equation 63
14.6. Representations of the Quantum Algebra Uq(sl2) and
^ Orthogonal Polynomials 64
14.6.1. Matrix elements of representations and
g Krawtchouk polynomials 64
14.6.2. The product and the addition theorems for
g Krawtchouk polynomials 65
14.6.3. A g analog of the Burchnall Chaundy formula 66
14.6.4. CGC s and ? Hahn polynomials 68
14.6.5. RC s and g Racah polynomials 70
14.6.6. The addition theorem for g Hahn polynomials 72
14.6.7. The addition formula for g Racah polynomials 76
14.6.8. RC s and properties of basic hypergeometric functions . 77
14.6.9. Relations for little g Jacobi polynomials and CGC s . . 79
14.7. g Askey Wilson Polynomials and their Special Cases ... 81
14.7.1. g Askey Wilson polynomials 81
14.7.2. Properties of q Askey Wilson polynomials 85
14.7.3. g Gegenbauer polynomials 87
14.7.4. Continuous g Hermite polynomials 90
14.7.5. Continuous 5 Jacobi polynomials 91
14.7.6. Big g Jacobi polynomials 94
14.8. Analysis on the Quantum Group SLq(2, C) and
Little g Jacobi Polynomials 97
14.8.1. The algebra of functions on the quantum group SLq(2, C) 97
14.8.2. Decomposition of the Hopf algebra A 99
Table of Contents ix
14.8.3. Finite dimensional corepresentations of A 101
14.8.4. Calculation of matrix elements 103
14.8.5. Irreducibility of the representations T( 105
14.8.6. Invariant integral on A 106
14.8.7. Scalar products on A(SUq(2)) 108
14.8.8. Unitary representations of the quantum group SUq{2) . 110
14.8.9. An analog of the Peter Weyl theorem 112
14.8.10. The Fourier transform on the quantum group SUq(2) . 113
14.8.11. Orthogonality of little g Jacobi polynomials.
g Legendre and Wall polynomials 114
14.8.12. Addition and product formulas for g Legendre
polynomials 116
14.8.13. The differential form of the quantum group SLq(2,C) . 119
14.8.14. The differential form of corepresentations 120
14.8.15. The difference equation for little j Jacobi polynomials . 122
14.8.16. The Rodrigues formula for little ? Jacobi polynomials . 123
14.9. Representations of the Quantum Group SUq(2) on
Quantum Spheres and g Orthogonal Polynomials 124
14.9.1. The algebra of functions on a quantum 2 sphere .... 124
14.9.2. Decomposition of the algebra A(Sq) 126
14.9.3. An invariant integral in Sq 127
14.9.4. Spherical functions on A(S%) 128
14.9.5. The orthogonality relation 130
14.9.6. The difference equation 131
14.9.7. The algebra of functions on a quantum 3 sphere .... 132
14.9.8. Spherical functions on Sq and big
g Jacobi polynomials 135
Chapter 15:
Semisimple Lie Groups and Related
Homogeneous Spaces 137
15.1. Decompositions of Semisimple Lie Algebras and Groups . 137
15.1.1. Decompositions of s[(n,C) and SL(n,C) 137
15.1.2. Cartan subgroups and subalgebras. Roots and
root subspaces 139
15.1.3. Generating elements of complex semisimple Lie algebras 150
15.1.4. Restricted roots and root subspaces 152
15.1.5. Real simple Lie groups and algebras 153
15.1.6. The Iwasawa decomposition 154
15.1.7. The Gauss decomposition 158
x Table of Contents
15.1.8. The Bruhat decomposition 159
15.1.9. The Cartan decomposition 161
15.1.10. Decompositions of classical groups 164
15.1.11. Noncompact analogues of the Iwasawa and
Cartan decompositions 167
15.1.12. Block (partial) decompositions of groups and
parabolic subgroups 168
15.1.13. Limits and contractions of Lie algebras 169
15.2. Homogeneous Spaces with Semisimple Motion Groups . . 172
15.2.1. Homogeneous self adjoint cones 172
15.2.2. Hermitian symmetric space 174
15.2.3. Tube domains 179
15.2.4. Parametrizations of the space ^Jm(F) 181
15.2.5. Spherical, conic and flag spaces 183
15.3. Invariant Metrics, Measures, and Differential Operators
on Lie Groups and on Homogeneous Spaces 185
15.3.1. Relations between invariant measures on Lie groups . . 185
15.3.2. Invariant metrics and measures on homogeneous cones . 190
15.3.3. Laplace operators on semisimple Lie groups and
their radial parts 194
Chapter 16:
Representations of Semisimple Lie Groups and
Their Matrix Elements 199
16.1. Irreducible Finite Dimensional Representations of
Lie Groups 199
16.1.1. Representations of Lie groups with normal
Gauss decompositions 199
16.1.2. Finite dimensional irreducible representations
of classical complex Lie groups 201
16.1.3. Block Gauss decompositions and representations .... 204
16.1.4. The Kostant theorem on separation of variables .... 205
16.1.5. Realization of finite dimensional representations on
spaces of polynomials in minors 208
16.1.6. Decomposition of symmetric powers of finite
dimensional irreducible representations 211
16.1.7. Restrictions of irreducible representations of
classical groups 214
16.1.8. The scalar product in the space V(VJlmn(F)) 216
Table of Contents xi
16.2. The Principal Series Representations of Classical Lie
Groups and Their Matrix Elements 217
16.2.1. The principal series representations of
the group GL(n, C) 217
16.2.2. Representations of real semisimple Lie groups 220
16.2.3. Realization of the principal series representations in
spaces of functions on matrix cones and hyperboloids 225
16.2.4. Relations between finite and infinite dimensional
representations of classical groups 228
16.2.5. Representations of nilpotent groups and of
semidirect products 229
16.2.6. Block splittings of matrices of irreducible
representations 232
16.2.7. The orthogonality relation for rows and columns .... 235
16.2.8. Block matrix elements of irreducible representations of
semisimple Lie groups 238
16.2.9. Integral expression for matrix elements of the
principal series representations 239
16.3. Hypergeometric Functions of Many Variables and
Representations of the Group GL(n, R) 240
16.3.1. The Lauricella functions 240
16.3.2. The most degenerate series representations of
the group SL(n, R) 244
16.3.3. Generalized beta functions and the
kernels K B 245
16.3.4. The Lauricella functions and the kernels K B 247
Chapter 17:
Group Representations and Special Functions of
a Matrix Argument 251
17.1. Elementary Functions of a Matrix Argument.
Gamma Function and Beta Function 251
17.1.1. Elementary functions of a matrix argument 251
17.1.2. The Fourier and the Laplace transforms of
functions of a matrix argument 252
17.1.3. The Fourier transform of harmonic polynomials .... 255
17.1.4. Generalized gamma functions 257
17.1.5. Generalized beta functions 260
17.1.6. Matrix analogues of the integral / (1 + x2)~adx ... 262
— oo
xii Table of Contents
1
17.1.7. Matrix analogues of the integral J(l — x2)xdx 265
o
17.2. Zonal Spherical Functions and Characters 270
17.2.1. Gel fand pairs 270
17.2.2. Zonal spherical functions and their properties 272
17.2.3. Characters of representations as spherical functions . . 275
17.2.4. Evaluation of characters of irreducible representations of
classical Lie groups 276
17.2.5. Identities for characters of irreducible
representations of GL(n, C) 280
17.2.6. Evaluation of zonal spherical functions of
classical complex Lie groups 285
17.2.7. The Green functions 286
17.2.8. Spherical transforms 288
17.2.9. Average values and Laplace operators 290
17.2.10. The algebra of representations 291
17.3. Zonal and Intertwining Polynomials 295
17.3.1. Recurrence formulas 295
17.3.2. Spherical functions as orthogonal polynomials 298
17.3.3. Invariant polynomials 300
17.3.4. Zonal spherical polynomials and their properties .... 302
17.3.5. Integral representations of zonal spherical polynomials . 305
17.3.6. The Laplace transform of zonal polynomials 305
17.3.7. Evaluation of zonal spherical polynomials 307
17.3.8. Intertwining functions 309
17.3.9 Generalized Jacobi polynomials 313
17.3.10. Generalized Jacobi polynomials and intertwining
operators 317
17.3.11. Zonal spherical functions and generalized
Jacobi and Bessel functions 318
17.3.12. Generalized Gel fand pairs 321
17.3.13. Ordered symmetric spaces and Volterra algebras .... 322
17.3.14. Zonal spherical functions on the space ^Jpg(F) 324
17.4. Hypergeometric Functions of a Matrix Argument .... 328
17.4.1. Hypergeometric functions on .f)m(F) 328
17.4.2. Bessel functions of a matrix argument 331
17.4.3. Hankel transforms of functions of a matrix argument . . 333
17.4.4. Bessel functions of the second kind in a matrix argument 340
17.4.5. Macdonald functions of a matrix argument 343
Table of Contents xiii
17.4.6. The confluent hypergeometric function of
a matrix argument 346
17.4.7. Whittaker functions of a matrix argument 348
17.4.8. Generalized Laguerre polynomials 351
17.4.9. The Gauss hypergeometric function of a matrix argument 354
17.4.10. Jacobi and Gegenbauer functions of a matrix
argument 357
Chapter 18:
Representations in the Gel fand Tsetlin Basis and
Special Functions 361
18.1. Infinitesimal Operators of Representations of
the Groups U(n) and S0(n) 361
18.1.1. The Gel fand Tsetlin basis 361
18.1.2. Infinitesimal operators of irreducible representations . . 362
18.2. Clebsch Gordan Coefficients for the
Gel fand Tsetlin Basis 365
18.2.1. Definition of Clebsch Gordan coefficients 365
18.2.2. Scalar factors 367
18.2.3. Tensor operators 370
18.2.4. The Wigner Eckart theorem 371
18.2.5. Matrix elements of the operators £*_! n and E1* n_1
of representations of g[(n,C) 373
18.2.6. CGC s for the tensor product Tmn ® T(p 0) 376
18.2.7. Evaluation of scalar factors 378
18.2.8. CGC s of the tensor product Tm g T(Oi_p) 381
18.2.9. CGC s of the tensor product of Tmn with
fundamental representations 382
18.2.10. CGC sof the tensor product Tmn ®T(1?0) 384
18.3. Matrix Elements of Representations of the Group GL(n,C)
and General Beta Functions 388
18.3.1. Matrix elements of irreducible finite dimensional
representations of GL(n,C) 388
18.3.2. General beta functions, related to the
Gel fand Tsetlin basis 389
18.3.3. Matrix beta functions 391
18.3.4. Recurrence formulas for general beta functions .... 393
18.4. Representations of U(n) in the Gel fand Tsetlin Bases
and Special Functions 395
18.4.1. Matrix elements of the representations of the group U(n) 395
xiv Table of Contents
18.4.2. The symmetry relations 397
18.4.3. Matrix elements of the fundamental representations . . 400
18.4.4. Matrix elements and CGC s 400
18.4.5. Matrix elements of representations of U(n) and
generalizations of classical polynomials of
a discrete variable 403
18.4.6. Representations of U(n) and generalized
Jacobi polynomials 403
18.4.7. The addition theorem for the polynomials Fi and F2 . ¦ 406
18.4.8. Recurrence relations 406
18.4.9. Orthogonality relations 408
18.5. Matrix Elements of Representations of the Groups
U(n 1,1), IU(n l) in the Gel fand Tsetlin Basis 409
18.5.1. Representations of the group U(n 1,1) 409
18.5.2. Matrix elements of the representations of U(n — 1,1) . . 411
18.5.3. Representations of the group IU(n — 1) 413
18.5.4. Matrix elements of the representations of IU(n — 1) . . 414
18.6. Representations of the Groups SO(n), SO0(n — 1,1),
ISO(n — 1) and Special Functions with Matrix Indices . . 416
18.6.1. Introduction 416
18.6.2. Representations of the groups SOq{u — 1,1)
and JSO(n l) 417
18.6.3. Matroms of representations 421
18.6.4. Differential relations of the first order 422
18.6.5. Differential equations of the second order 425
18.6.6. Bessel and Jacobi functions with matrix indices .... 427
18.7 Orthogonal Polynomials of Many Discrete
and Continuous Variables 432
18.7.1. General procedure of spectral analysis of infinitesimal
operators and matroms 432
18.7.2. Partial difference equations connected with infinitesimal
operators and matroms 433
18.7.3. Spectral characteristics of discrete equations 438
18.7.4. Continuous analogs of discrete polynomials 440
18.7.5. Expansions of representation matrix elements 444
Chapter 19:
Modular Forms, Theta Functions and Representations of
Affine Lie Algebras 447
19.1. Modular Forms 447
Table of Contents xv
19.1.1. Linear fractional transformations of the upper half plane 447
19.1.2. The transformation group SL(2, Z) 449
19.1.3. Congruence subgroups of SL(2, Z) 451
19.1.4. Modular forms of an integral weight 453
19.1.5. Eisenstein series 456
19.1.6. Modular forms with multiplier system 459
19.2. Theta Functions 462
19.2.1. The Jacobi theta functions 462
19.2.2. Functional equation 465
19.2.3. The Jacobi theta functions and the heat equation . . . 470
19.2.4. Factorization of the theta function into infinite product 471
19.2.5. Theta functions with characteristics 473
19.2.6. Theta functions of many variables 476
19.2.7. The symplectic group 479
19.2.8. The functional equation for the theta function of
many variables 480
19.2.9. Theta functions of many variables with characteristics . 483
19.2.10. Relations for products of theta functions 484
19.3. Theta Functions and the Decomposition of Quasi Regular
Representation of the Heisenberg Group on the Cube . . 486
19.3.1. Auxiliary theta functions 486
19.3.2. The space #n(a/n) 491
19.3.3. Decomposition of the quasi regular representation . . . 494
19.3.4. The orthonormal basis of the space Hn(a/n) 497
19.4. Afflne Lie Algebras 498
19.4.1. Non twisted affine Lie algebras 498
19.4.2. Roots and root elements of non twisted affine
Lie algebras 500
19.4.3. The Virasoro algebra 505
19.4.4. The affine Lie algebra A[^ 506
19.4.5. Twisted affine Lie algebras 509
19.4.6. The affine Lie algebra A{22) 515
19.4.7. Classification of affine Lie algebras 519
19.4.8. The universal enveloping algebra 524
19.5. Representations of Afflne Lie Algebras and
their Characters 526
19.5.1. Integrable and weight representations 526
19.5.2. Verma modules 527
19.5.3. Integrable representations with highest weight 529
xvi Table of Contents
19.5.4. Characters of integrable representations 530
19.6. Characters of Representations of the Afflne Lie Algebras
and Combinatorial Identities 534
19.6.1. The denominator formula for the algebras A ,
A2 and the Jacobi identity 534
19.6.2. Specialized characters of the algebra A 536
( 2)
19.6.3. Specialized characters of the algebra A 540
19.7. Characters of Representations and Theta Functions . . . 542
19.7.1. An other form of theta functions 542
19.7.2. The lattices M and M 546
19.7.3. Maximal weights of irreducible integrable representations 552
19.7.4. Characters of representations and theta functions . . . 553
19.7.5. The functions Ax and A x 557
19.7.6. Expressions for the function Ap 561
19.7.7. Transformation properties of the function Ap 563
19.7.8. Polynomial algebras 568
19.8. The String Function 569
19.8.1. Properties of the string function 569
19.8.2. The matrix of the string functions 572
19.8.3. Explicit expressions for the string functions 575
19.8.4. Formulas for the partition function 579
19.8.5. Hecke modular forms and the string function forAj . . 582
19.8.6. Applications of the string functions 588
19.9. Reduction of Representations of an Afflne Lie Algebra onto
a Subalgebra and Hecke Modular Forms 590
19.9.1. The functions Eejk 590
19.9.2. The matrix (e fc) 593
19.9.3. Evaluation of Eejk 594
19.9.4. Reduction C^ D C{t1] 600
Bibliography 603
Bibliography Notes 621
Subject Index 625
|
| any_adam_object | 1 |
| author | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 |
| author_GND | (DE-588)127328122 (DE-588)115774580 |
| author_facet | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 |
| author_role | aut aut |
| author_sort | Vilenkin, Naum Ja. 1920-1991 |
| author_variant | n j v nj njv a u k au auk |
| building | Verbundindex |
| bvnumber | BV005847546 |
| ctrlnum | (OCoLC)312016259 (DE-599)BVBBV005847546 |
| format | Book |
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| id | DE-604.BV005847546 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:35:43Z |
| institution | BVB |
| isbn | 079231493X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003661615 |
| oclc_num | 312016259 |
| open_access_boolean | |
| owner | DE-384 DE-91G DE-BY-TUM DE-12 DE-739 DE-824 DE-29T DE-188 |
| owner_facet | DE-384 DE-91G DE-BY-TUM DE-12 DE-739 DE-824 DE-29T DE-188 |
| physical | XIX, 629 Seiten |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Kluwer Academic Publications |
| record_format | marc |
| series | Mathematics and its Applications. Soviet Series |
| series2 | Mathematics and its Applications. Soviet Series Mathematics and its Applications / Soviet Series |
| spelling | Vilenkin, Naum Ja. 1920-1991 Verfasser (DE-588)127328122 aut Representation of Lie groups and special functions 3 Classical and quantum groups and special functions by N. Ja. Vilenkin and A. U. Klimyk Dordrecht [u.a.] Kluwer Academic Publications 1992 XIX, 629 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematics and its Applications. Soviet Series 75 Mathematics and its Applications / Soviet Series ... Klimyk, Anatolij U. 1939-2008 Verfasser (DE-588)115774580 aut (DE-604)BV005437238 3 Mathematics and its Applications. Soviet Series 75 (DE-604)BV004708148 75 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003661615&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 Representation of Lie groups and special functions Mathematics and its Applications. Soviet Series |
| title | Representation of Lie groups and special functions |
| title_auth | Representation of Lie groups and special functions |
| title_exact_search | Representation of Lie groups and special functions |
| title_full | Representation of Lie groups and special functions 3 Classical and quantum groups and special functions by N. Ja. Vilenkin and A. U. Klimyk |
| title_fullStr | Representation of Lie groups and special functions 3 Classical and quantum groups and special functions by N. Ja. Vilenkin and A. U. Klimyk |
| title_full_unstemmed | Representation of Lie groups and special functions 3 Classical and quantum groups and special functions by N. Ja. Vilenkin and A. U. Klimyk |
| title_short | Representation of Lie groups and special functions |
| title_sort | representation of lie groups and special functions classical and quantum groups and special functions |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003661615&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005437238 (DE-604)BV004708148 |
| work_keys_str_mv | AT vilenkinnaumja representationofliegroupsandspecialfunctions3 AT klimykanatoliju representationofliegroupsandspecialfunctions3 |