Lie Theory and Its Applications in Physics Ixth International Workshop

Cover......Page 1
Lie Theory and Its Applications in Physics......Page 4
Preface......Page 6
Contents......Page 8
List of Participants......Page 12
1 Introduction......Page 16
2 The Analytic Tachyon Vacuum Solution......Page 17
3.1 Analytic Lump Solutions......Page 19
References......Page 24
1 Introduction......Page 26
2.1 p-Adic Numbers and Their Applicability......Page 27
2.2 Scattering Amplitudes and Lagrangian for Open Scalar p-Adic Strings......Page 29
2.3 New Kind of Matter, Which Has Origin in p-Adic Strings......Page 30
3 Adelic Cosmological Modelling......Page 32
3.1 Adeles and Their Applicability......Page 33
3.2 Some Adelic Cosmological Investigations......Page 34
4 Concluding Remarks......Page 35
References......Page 36
1 Modular Architecture......Page 38
2 Palev Statistics......Page 39
2.2 Di-Fermions......Page 40
2.4 Feynman Space-Time......Page 41
2.6 Regular Space-Time-Momentum-Energy......Page 42
2.7 Interpretation of the Indefinite Norm......Page 43
2.9 Quantification and Quantization......Page 44
2.10 Gauge......Page 45
3 Higher-Order Quantum Set Theory......Page 46
3.1 Spin Structure of S"0362S......Page 47
4.2 Constituents of the Photon......Page 49
References......Page 50
1 Introduction......Page 52
2 Weierstrass Canonical Form for Elliptic Curves......Page 55
3 A Joke and a Misunderstanding......Page 62
4 The Hessenberg and Its Generalizations......Page 66
References......Page 72
1 Introduction......Page 74
2 Preliminaries......Page 76
3 Macdonald's Formula......Page 79
4 Derivation of the LSV Result......Page 81
5 Classical Group Characters......Page 82
6 Howe Dual Pairs......Page 84
7 Metaplectic Representations......Page 86
8 Conclusions......Page 87
References......Page 88
1 What are Minimal Representations?......Page 89
2 More Symmetric Than Symmetric Spaces......Page 91
3 Schrödinger Model of Minimal Representations......Page 92
4 Special Functions to 4th Order Differential Operators......Page 94
5 Broken Symmetries and Branching Laws......Page 95
6 Generalized Fourier Transform as a Unitary Inversion......Page 97
7 SL2-Triple in the Schrödinger Model......Page 98
8 Quantization of Kostant–Sekiguchi Correspondence......Page 101
References......Page 103
1 Introduction......Page 106
2.2 Free Zinbiel Algebra......Page 108
2.5 Dendriform Algebra......Page 109
2.8 Dendriform Calculus......Page 110
2.10 A Commutative Diagram of Algebras......Page 112
3.1 Exponential Series Without Denominators......Page 113
3.4 Zinbiel Algebras in Characteristic p......Page 114
4 BCH-Type Formula for the Exponential Series Without Denominators......Page 115
4.3 Functional Equation in One Variable......Page 116
4.5 The Multilinear Case......Page 117
References......Page 118
1 Introduction......Page 119
2 Higher Chiral Partial Waves......Page 121
3 Intertwining Differential Operators......Page 125
4 Application: Test of Positivity of a 6-Point Structure......Page 130
References......Page 135
Euclidean Configuration Space Renormalization, Residues and Dilation Anomaly......Page 136
1 Introduction......Page 137
2 General Requirements. Reduction to a One-Dimensional Problem......Page 141
3 Residues and Renormalization of Primitively Divergent Graphs......Page 144
4 Dilation Anomaly. Examples of Graphs with Subdivergences......Page 148
5 Concluding Remarks......Page 151
References......Page 154
1 Introduction and Some History......Page 157
2 Wigner Quantum Systems and Palev's Contributions......Page 160
3 Main Example: The n-Dimensional Non-isotropic Harmonic Oscillator......Page 163
4 Recent Advances: Angular Momentum Operators and Their Spectrum......Page 168
References......Page 171
1 Introduction and Discussion......Page 174
2 SUSY Quantum Mechanics: SUSY Breaking and External Field......Page 176
3 Localization in SUSY Matrix Models......Page 177
3.1.1 Localization to W'(φ)=0, and Localization Versus Vandermonde......Page 179
4 (eiα-1)-Expansion and Nicolai Mapping......Page 181
5.1 General Two-Cut Solutions......Page 182
5.2 Symmetric One-Cut Solution......Page 183
References......Page 184
1 Introduction......Page 185
2 CPN-1 Case......Page 188
References......Page 193
1 Introduction......Page 194
2 Operadic Construction of the Group of Formal Diffeomorphisms......Page 195
3 Group Associated to a Symmetric Operad......Page 197
4 Feynman Diagrams and Their Combinatorics......Page 200
5 The Universal Contraction Operad......Page 204
6 Suboperads in R and Concrete Combinatorial Models of Quantum Field Theory......Page 206
7.1 Notions of Renormalization Group......Page 208
7.3 The Wick Generating Operator of Diagrams......Page 211
7.4 Construction of Operadic Morphism Ξ : RE,V → End KV......Page 212
References......Page 213
1 Introduction......Page 215
2 Lagrangian Formulation of Lightlike Brane Dynamics......Page 217
3 Gravity/Gauge-Field System Interacting with Lightlike Brane......Page 219
4 Explicit Solutions: Braneworlds via Lightlike Brane......Page 222
5 Traversability and Trapping Near the Lightlike Braneworld......Page 224
6 Discussion......Page 225
References......Page 228
Generalized Bernoulli Polynomials and the Casimir Effect in the Einstein Universe......Page 231
1 Preliminaries on the Einstein Universe and Minkowski Space in n Dimensions......Page 232
2 Quantization of a Real Massless Scalar Field on......Page 233
3 The Renormalized Vacuum Energy of a Massless Scalar Field on  and the Casimir Energy of a Massless Scalar Field on  with Dirichlet Boundary Conditions on Σn-1......Page 234
4 Generalized Bernoulli Polynomials and the Casimir Energy......Page 236
References......Page 238
1 Introduction and Motivation......Page 239
2 The Commutator-Intersection Relations (CIRs)......Page 240
3 Lie Algebras of the Third Kind......Page 242
4 Example of a LATKe......Page 243
5 Classification of Simple LATKes......Page 245
6.1 LATKe Gauge Theory......Page 246
6.2 Uniqueness of the LATKe as a Vacuum Selection Mechanism......Page 247
References......Page 248
1 Introduction......Page 250
2 On a Nonlocal Modification of Gravity......Page 251
2.1 Case  R = r R......Page 252
2.2 Case  R = q R2......Page 253
2.3 Case n R = cn Rα n + β......Page 254
3 Concluding Remarks......Page 257
References......Page 258
1 Introduction......Page 259
2 Superalgebra Uq(sl(N|1)) and Uq,p(sl(N|1))......Page 260
2.1 Quantum Superalgebra Uq(sl(N|1))......Page 261
2.2 Elliptic Superalgebra Uq,p(sl(N|1))......Page 262
3.1 Boson......Page 264
3.2 Quantum Superalgebra Uq(sl(N|1))......Page 265
3.3 Elliptic Superalgebra Uq,p(sl(N|1))......Page 267
4 q-Wakimoto Realization......Page 268
References......Page 270
1 Introduction......Page 272
2 The Alternative θ-Deformation of Quantum Mechanics......Page 273
3 Quantum Phases in Noncommutative Space......Page 274
References......Page 280
1 Introduction......Page 282
2 Uq in Disguise: The FRT Construction......Page 284
3 The Diagonal Monodromy Matrix Mp......Page 288
4 The Quantum Determinant detq (M)......Page 290
References......Page 291
1 Introduction......Page 293
2 Review of Galilean Conformal Algebras in (d+1)D Spacetime......Page 294
3 Centrally Extended Algebra and Its Verma Modules......Page 295
4 Coadjoint Representation of  g......Page 298
5 Concluding Remarks......Page 300
References......Page 301
1 Introduction......Page 302
2 Preliminaries......Page 303
3 The Non-compact Lie Algebras sp(n,IR)......Page 305
4.1.1 Main Multiplets......Page 308
4.1.2 Reduced Multiplets R61......Page 310
4.1.3 Reduced Multiplets R62......Page 311
4.1.4 Reduced Multiplets R63......Page 312
4.1.5 Reduced Multiplets R64......Page 313
4.1.6 Reduced Multiplets R65......Page 314
4.2 The Cases sp(n,IR) for n5......Page 315
5 Outlook......Page 316
6 Figures......Page 317
References......Page 325
1 Introduction......Page 327
2 Generalized Formula......Page 329
3 Gauge Affine Action......Page 332
4 Gauge Affine Symmetry Vertex Coefficients Evaluation......Page 335
References......Page 336
1 Introduction......Page 338
2.1 Algebraic Structure......Page 339
2.2 Representation Theory......Page 340
2.3 The Representation Ring......Page 341
3.1 Algebraic Structure......Page 342
3.2 Representation Theory......Page 343
3.3 Fusion......Page 344
4.2 Characters of Extended Algebras......Page 346
4.3 Free Field Realisations......Page 347
4.4 The Extended Operator Product Algebra......Page 348
4.5 Examples......Page 350
4.6 W(2)N-Subalgebras......Page 353
References......Page 355
1 Introduction......Page 357
2 Non-Local Representation of the Ageing Algebra age(1)......Page 359
3 Finite Transformations......Page 360
4 Covariant Two-Point Functions......Page 363
References......Page 366
1 Introduction......Page 368
2 The SU(m,1) Group and Its Lie Algebra......Page 369
3 The Holomorphic Discrete Series of Representationsof the SU(m,1) Group......Page 371
4 The Coherent States Quantization of D=SU(2,1)/U(2) and the Star Product......Page 372
References......Page 374
1 Introduction......Page 376
2 N0 Modulo C2(N0)......Page 377
3 N0 Modulo C2((V(k,0))......Page 378
4 Singular Vectors and Ideals of AA......Page 381
References......Page 383
1 Introduction......Page 384
2 Super Vertex Algebras and Boson-Fermion Correspondence of Type A......Page 386
3 Twisted Vertex Algebras and Boson-Fermion Correspondence of Type B......Page 390
References......Page 394
1 Introduction......Page 396
2 The N=1 and N=2 Superconformal Algebras and Their Virasoro-Preserving Automorphisms......Page 398
3 The Notions of VOSA, Supersymmetric VOSA and Twisted Module......Page 400
4 Twisting by Automorphisms Arising from Virasoro-Preserving Automorphisms of the N=2 Neveu–Schwarz Algebra......Page 402
References......Page 404
1 Introduction......Page 406
2 The Concept of Ruijsenaars Duality......Page 407
3.1 Quasi-Hamiltonian Systems on D and Their Reductions......Page 409
3.2 Symplectic Action of the Mapping Class Group on P(0)......Page 411
4 Compactified Ruijsenaars–Schneider IIIb System......Page 412
5 Self-Duality of the IIIb System from Reduction......Page 415
6 Further Results and Open Problems......Page 418
References......Page 420
1 The FQHE Fabry–Pérot Interferometer......Page 421
2 Coulomb Blockade Island'S Conductance: The CFT Point of View......Page 423
2.2 Coulomb Island'S Conductance......Page 424
3 Decomposable Subalgebra and ZnH Grading......Page 425
4 The RCFT Partition Function for a General FQH Disk......Page 429
5 Application: Coulomb Blockade in the Z3 Read–Rezayi State......Page 430
References......Page 431
1 Introduction......Page 433
2 Equivalence Transformations......Page 434
3 Lie Symmetries......Page 437
4 Generation of Exact Solutions......Page 439
5 Conclusion......Page 440
References......Page 441
1 Introduction......Page 442
3 Potential Representation of Vector Quantities of Magnetohydrodynamics......Page 444
4 The Action of Barotropic Magnetohydrodynamics......Page 445
5 A Simpler Action for Barotropic Magnetohydrodynamics......Page 446
6 Diffeomorphism Symmetry and Noether Currents......Page 447
References......Page 449
1 Introduction......Page 450
3 Exceptional Quantum Mechanics (F4)......Page 451
4 Complex Jordan Algebras (E6)......Page 454
References......Page 456
1 Introduction......Page 457
2 The Spectral Problem......Page 458
4 The Determining Equations......Page 460
5 Superpotentials......Page 461
References......Page 462
1 Introduction......Page 464
2 The Case of Defect One......Page 466
3 The Case of gl(2|2)......Page 468
4 The Case of gl(n|n)......Page 470
5 Truncated Super-Yangians......Page 473
References......Page 474
1 Introduction......Page 475
2 Artin–Schelter Regularity......Page 476
3 Homology and Cohomology of g......Page 478
4 Littlewood Formula and PS......Page 479
5 Homotopy Algebras A∞ and C∞......Page 480
6 Homotopy Transfer Theorem......Page 481
References......Page 485
1 Introduction......Page 486
2 Modular Data......Page 487
3.1 The A-Series......Page 490
4 The NIM-rep......Page 491
5 Concluding Remarks......Page 493
References......Page 494
1 Introduction and Preliminaries......Page 495
2 Differential Invariants......Page 496
3 The Chern Connection Attached to a SODE......Page 497
4 Statement of the Main Result......Page 499
5 Concluding Remarks......Page 502
References......Page 503
1 Introduction......Page 504
2 Definitions of Harmonic and Quasi-Conformal Mappings......Page 505
3 Ahlfors–Schwarz Lemma......Page 507
4 Ahlfors–Schwarz Lemma for Harmonic Quasi-Conformal Mappings......Page 510
References......Page 512
1 Introduction......Page 513
2 Lie Algebras in Additive Monoidal Categories......Page 514
3 Functorial Properties......Page 518
References......Page 521
1 Introduction......Page 523
3 The Continuous Bialgebra Structure......Page 524
References......Page 526