Lie Groups: New Research (Mathematics Research Developments Series)

LIE GROUPS: NEW RESEARCH......Page 4
CONTENTS......Page 6
PREFACE......Page 8
Introduction......Page 16
The Clifford Algebra Cl3,0 of Euclidean 3-Space......Page 19
New Review of Angular Momentum Algebra......Page 20
Quantum Spacetime......Page 25
The CNC Duality in Quantum Geometry......Page 27
Lie Groups in Clifford Quantum Geometry......Page 28
Maximal Cartan Subalgebra in Lp,q......Page 31
The Sakata Model......Page 32
The Spacetime Algebra Cl3,1 and its Lie group L3,1......Page 33
The Spacetime Lie Algebra L(2)......Page 38
Pure Space and Constitutive Group......Page 39
Pure space of Clifford Algebra Cl3,1......Page 41
Some SU(3) History......Page 42
The Spacetime Oscillator......Page 44
The Spacetime Color Degree of Freedom......Page 48
CPT and Spacetime Area Oscillation......Page 49
Out of the Light . . .......Page 50
Into the Dark......Page 54
Towards a New Concept of Motion: Appendix A......Page 60
Relativity Revisited: Appendix B......Page 65
References......Page 70
1. Introduction......Page 76
2. Manifolds and Their Tangents......Page 78
3. Rotation Manifold and Its Tangents......Page 80
4. Angular Velocities, Accelerations Vectors......Page 91
5. Euler’s Equations......Page 94
6. Simplified Newmark Integration Scheme......Page 96
7. Interpolation of the Rotation Field and Its Objectivity......Page 98
Conclusions......Page 101
References......Page 102
1.Introduction......Page 104
2.Iwasawa Coordinates and Metrics on G=PSL2(R)......Page 107
3.Geometry of a Modular Homogeneous Space......Page 110
4.Closed Forms and Harmonic Forms......Page 114
5.Left Brownian Motion on G=PSL2(R)......Page 117
6.Asymptotic Brownian Windings in......Page 118
7.Geodesics of G=PSL2(R) and Ergodic Measures......Page 119
8.Asymptotic Geodesic Windings......Page 123
9.Equirepartition in of Large Geodesic Spheres......Page 125
10.Synthetic Proof of Theorem 6.1......Page 127
11.Proof of Theorem 8.1......Page 131
References......Page 135
1.Introduction......Page 138
2.Reproduction of Solutions......Page 141
3.Ideal Plane Plasticity System and Its Known Solutions......Page 142
4.Transformation of Solutions......Page 146
References......Page 152
1.Introduction......Page 154
2.The Module Structure of Vect(M)......Page 157
3.The Module Structure of L(X)......Page 159
4.Some Applications......Page 171
5.Conclusion......Page 180
References......Page 181
Abstract......Page 184
2.Lie Group Methods......Page 185
3.Simulations of Differential Equations by RKMK Method......Page 190
4.Simulations of Modulus Conserving Differential Equations by Magnus Method......Page 196
References......Page 199
Abstract......Page 202
1.Some Preliminaries......Page 203
2.Singularities and Stability of a Work Function......Page 206
3.Examples......Page 211
References......Page 213
Abstract......Page 214
1.Introduction......Page 215
2.The Conformal-Affine Structure......Page 219
3.The Physical Realization......Page 242
4.Applications and Consequences in open Quantum Relativity......Page 262
5.Discussion, Conclusions and Perspectives......Page 270
6.Appendix......Page 273
References......Page 277
Abstract......Page 282
1.Twisted Balanced Metrics......Page 283
2.The Moment Map Picture......Page 287
3.Approximation of Twisted cscK Metrics......Page 289
4.Further Directions......Page 294
References......Page 295
Abstract......Page 298
1.Introduction......Page 299
2.Reduction of a Geodesic Flow to a Lie Group......Page 301
3.Ideal Flows on Lie Groups......Page 305
4.Controllability, Bernoulli Functions and Flows with External Friction......Page 307
5.n-dimensional Euler Top......Page 309
6.Reduction of the Euler Equations for Ideal Incompressible Fluid......Page 312
7.Symmetry Fields and Invariants of Coadjoint Orbits......Page 315
8.Conclusion......Page 317
A. Reduction to the Lie Group for Nonholonomic Systems......Page 318
B.Controllability Conditions......Page 319
References......Page 320
Abstract......Page 322
Introduction......Page 323
1. -scaled Vector Fields and <> Systems of Differential Equations......Page 324
2. Quasimetrics......Page 326
3. Local Homogeneous Approximations of Collections of Vector Fields......Page 330
4. Some Applications......Page 332
References......Page 338
1.Introduction......Page 340
2.Basic Notions on Symplectic Manifolds......Page 342
3.Symplectic Actions: The AKS-Scheme......Page 348
4.Applications of the Adler-Kostant-Symes-scheme to Classical Mechanics......Page 352
5.Quadratic Hamiltonians and Coadjoint Orbits......Page 356
References......Page 362
Abstract......Page 366
1.An Informal Review of Lie Groups and Homogeneous Spaces......Page 367
2.L´evy Processes in Lie Groups......Page 368
3.Generators and Stochastic Integral Equations......Page 369
4.L´evy Processes in Compact Lie Groups......Page 373
5.L´evy Processes in Homogeneous Spaces......Page 376
6.L´evy-Khinchin Formula......Page 378
7.Symmetric Spaces......Page 379
8.Limiting Properties on Noncompact Type Symmetric Spaces......Page 381
9.Dynamical Aspect......Page 383
10.Nonhomogeneous L´evy Processes in Lie Groups......Page 386
11.Nonhomogeneous L´evy Processes in Homogeneous Spaces......Page 388
12.A Decomposition of a Markov Process......Page 391
References......Page 396
1.Introduction......Page 400
2.Case 1: A Nonlinear ODE......Page 401
3.Case 2: A Linear PDE: The Schr¨odinger Equation......Page 405
4.On the Reduction Techniques......Page 409
5.Conclusion......Page 411
References......Page 412
1.Introduction......Page 416
2.Levi Decomposition of Lie Algebras......Page 417
3.Deformations and Xohomology of Lie Algebras......Page 418
4.Contractions of Lie Algebras......Page 423
5.Contractions of Simple Lie Algebras......Page 426
6.Detailed Analysis of Deformations and Contractions. The Lie Algebra L8,14......Page 428
7.Contractions and Deformations of Non-solvable Lie Algebras in Dimension n7......Page 432
8.Deformations of Indecomposable Eight Dimensional Algebras with Levi Subalgebra......Page 434
9.Deformations of Decomposable Eight Dimensional Algebras with Levi Subalgebra......Page 446
Acknowledgment......Page 453
References......Page 459
Introduction......Page 462
1.An Introduction to Orbifolds......Page 464
2.G-structures on Orbifolds......Page 469
3.Automorphisms of Finite Type G-structures on Orbifolds......Page 477
4.Classes of Affinely Connected Orbifolds......Page 485
References......Page 496
1.Introduction......Page 500
2.Wrap Groups of Fibers......Page 502
3.Structure of Wrap Groups......Page 517
4.Cohomologies of Wrap Groups......Page 529
5.CW-groups for Wrap Groups......Page 568
References......Page 573
1.Introduction......Page 578
2.Groups of Diffeomorphisms......Page 580
3.Projective Decomposition of Diffeomorphism Groups......Page 586
4.Example of the Group of Diffeomorphisms......Page 592
5.One-parameter Subgroups of Diffeomorphism Groups......Page 595
6.Topological Perfectness of Diffeomorphism Groups......Page 600
7.Projective Decomposition of Wrap Groups......Page 602
8.Appendix......Page 608
References......Page 613
INDEX......Page 616