Noncompact Lie Groups and Some of Their Applications

During the past two decades representations of noncompact Lie groups and Lie algebras have been studied extensively, and their application to other branches of mathematics and to physical sciences has increased enormously. Several theorems which were proved in the abstract now carry definite mathematical and physical sig­ nificance. Several physical observations which were not understood before are now explained in terms of models based on new group-theoretical structures such as dy­ namical groups and Lie supergroups. The workshop was designed to bring together those mathematicians and mathematical physicists who are actively working in this broad spectrum of research and to provide them with the opportunity to present their recent results and to discuss the challenges facing them in the many problems that remain. The objective of the workshop was indeed well achieved. This book contains 31 lectures presented by invited participants attending the NATO Advanced Research Workshop held in San Antonio, Texas, during the week of January 3-8, 1993. The introductory article by the editors provides a brief review of the concepts underlying these lectures (cited by author [*]) and mentions some of their applications. The articles in the book are grouped under the following general headings: Lie groups and Lie algebras, Lie superalgebras and Lie supergroups, and Quantum groups, and are arranged in the order in which they are cited in the introductory article. We are very thankful to Dr.

1. Noncompact Lie groups, their algebras and some of their applications.- Lie Groups and Lie Algebras.- 2. Harish-Chandra’s c-function. A mathematical jewel.- 3. Basic harmonic analysis on pseudo-Riemannian symmetric spaces.- 4. The extensions of space-time. Physics in the 8-dimensional homogeneous space D = SU(2, 2)/K.- 5. Ordinary — and momentum — space conformai compactifications: Some possible observable consequences.- 6. Radon transform on halfplanes via group theory.- 7. Analytic torsion and automorphic forms.- 8. Diffusion on compact ultrametric spaces.- 9. Generalized square integrability and coherent states.- 10. Maximal abelian subgroups of SU(p, q) and integrable Hamiltonian systems.- 11. Path integrals and Lie groups.- 12. Representations of diffeomorphism groups and the infinite symmetric group.- 13. Characters of Lie groups.- 14. Weyl group actions on Lagrangian cycles and Rossmann’s formula.- 15. Taylor formula, tensor products, and unitarizability.- 16. A connection between Lie algebra roots and weights and the Fock space construction.- 17. Applications of Sp(3,R) in nuclear physics.- 18. Nilpotent groups and anharmonic oscillators.- 19. Extensions of the mass O helicity O representation of the Poincare group.- 20. Invariant causal propagators in conformai space.- 21. Gauge groups, anomalies and non-abelian cohomology.- 22. The E8 family of quasicrystals.- 23. Wavelet interpolation and approximate solutions of elliptic partial differential equations.- Lie Superalgebras and Lie Supergroups.- 24. From super Lie algebras to supergroups: Matrix realizations and the factorisation problem.- 25. Current algebras as Hilbert space operator cocycles.- 26. Non-linear realization technique — The most convenient way of deriving N = 1 supergravity.- 27. Todasystems as constrained linear systems.- Quantum Groups.- 28. On the definitions of the quantum group Uh(sl(2, k)) and the restricted dual of Uh(sl(n, k)).- 29. Universal T-matrix for twisted quantum gl(N).- 30. Unitary representations of quantum Lorentz group.- 31. Contraction of quantum groups and lattice physics.- 32. A quantum Poincaré group and the Dirac-Coulomb problem.