The Symposium ·Symmetries in Science VI: From the Rotation Group to Quantum Algebras· was held at the Cloister Mehrerau, Bregenz, Austria, during the period August 2-7, 1992. The Symposium was held in honor of Professor Lawrence C. Biedenharn on the occasion of his 70th birthday. During the academic year 1966/67 I worked as research associate with Larry at Duke University and we have ever since maintained close contact. It was thus natural for me to take the initiative and to organize this Symposium in honor of Larry as a great scientist and friend. The response which the Symposium received showed the favorable reaction by the scientific community to the opportunity provided by the Symposium to honor our colleague, teacher and friend. Equally, the scientific contributions contained in this volume illustrate the high esteem in which he is held. I wish to thank all the scientists who participated in the Symposium and who contributed to this volume. It is due to their commitment that the Symposium was successful. Finally I need to thank those who provided financial and logistical assistance to the Symposium: Dr. John H. Guyon, President of Southern Illinois University at Carbondale, Dr. Russell R. Dutcher, Dean, College of Science at SIUC, Dr. Maurice A. Wright, Chairman, Department of Physics, SIUC, Dr. Victoria J. Molfese, Office of Research Developement and Administration, SIUC, as well as Dr. Martin Purtscher, Landeshauptmann, Land Vorarlberg Dr. Guntram Lins, Landesrat, Land Vorarlberg.

The Symposium "Symmetries in Science VII: Spectrum Generating Algebras and Dynamic Symmetries in Physics" was held at the Southern Illinois University at Carbondale in Niigata, Japan Campus, during the period August 28-31, 1992. The Symposium was held in honor of Professor Francesco lachello on the occasion of his 50th birthday. We wish to thank the colleagues and friends of Franco for their participation in the Symposium as well as for contributing articles to this volume honoring him. It was their commitment and involvement which made this Symposium a success. We also wish to thank Dr. Jared H. Dorn, the director of SIUC-N, for his support in the planning and the execution of the Symposium. Moreover we wish to thank Mayor Nobuo Kumakura of Nakajo town and Mr. Kaichi Suzuki of the school entity "The Pacific" for their friendly support. Bruno Gruber, SIUC-N Takaharu Otsuka, University of Tokyo v LAUDATIO ON THE OCCASION OF THE 50TH BIRTHDAY OF PROFESSOR FRANCESCO IACHELLO I first met Franco lachello in 1974. Driving a smart Alfa-Romeo, he came to meet me at the station at Groningen where I was to spend a summer conducting research.

Thf3 symposium "Symmetries in Science VIII" was held in August of 1994 at the Cloister Mehrerau in Bregenz, Austria. The symposium was supported by Southern Illinois University at Carbondale, the Land Vorarlberg, and the Landeshaupstaot Bregenz. I wish to thank Dr. John C. Guyon, President of Southern Illinois University at Carbondale; Dr. Hubert Regner, Amt der Vorarlberger Landesregierung; and Dipl. Vw. Siegfried Gasser, Buergermeister der Landeshauptstadt Bregenz and Lantagsabgeordneter, for their generous support of the symposium. Finally I wish to thank Frater Albin of the Cloister Mehrerau for his support and cooperation in this endeavor, which made for a successful meeting in a most pleasant environment. Bruno Gruber v CONTENTS On Om x Gin Highest Weight Vectors Helmer Aslaksen, Eng-Chye Tan, and Chen-bo Zhu ... . Invariant Theory of Matrices Helmer Aslaksen, Eng-Chye Tan, and Chen-bo Zhu ... . 1 3 Symmetries of Elementary Particles Revisited A.O. Barut ... ... ... 21 Perturbative SU(1,1) Haluk Seker ... . ..., . ... 25 A Dual Structure for the Quantal Rotation Group, SU(2) L.C. Biedenharn and M.A. Lohe ..., . ... ... 37 Some Points in the Quantization of Relativistic Grassmann Dependent Interaction Systems A. Del Sol Mesa and R, P. Martinez y Romero ..., . ... 49 for Uq(sl(4)) and q-Conformal q-Difference Intertwining Operators I nvariant Equations V.K. Dobrev ... ... ... 55 A Quantum Mechanical Evolution Equation for Mixed States from Symmetry and Kinematics H.-D. Doebner and J.D. Hennig ... ... 85 vii Quantum Mechanical Motions over the Group Manifolds and Related Potentials I.H. Duru -. ... ... ...

Proceedings an International Symposium held in Bregenz, Austria, July 13-18, 1997

Proceedings of a symposium held in Landesbildungszentrum Schloss Hofen, Lochau, Vorarlberg, Austria, July 30-August 3, 1990

This volume contains material submitted to a symposium on mathematical physics in July 1993 at the Technical University of Clausthal. It focuses on mathematical and physical notions of symmetries, especially on generalized and quantum symmetries (quantum groups and algebras) with applications to integrable systems, quasi-exact solvability, quantization methods and nonlinear problems. The workshop is a continuation of the series of meetings and workshops on mathematical physics in Clausthal which brings together mathematicians, theoretical and mathematical physicists working in the different parts of generalized symmetries. Recent developments in the mathematical structure and physical applications of generalized symmetries are discussed.

Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics. This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator. Using this concept, properties of both the algebra and co-algebra are developed from a single uniform point of view, which is especially helpful for understanding the noncommuting co-ordinates of the quantum plane, which we interpret as elementary tensor operators. Representations of the q-deformed angular momentum group are discussed, including the case where q is a root of unity, and general results are obtained for all unitary quantum groups using the method of algebraic induction. Tensor operators are defined and discussed with examples, and a systematic treatment of the important (3j) series of operators is developed in detail. This book is a good reference for graduate students in physics and mathematics.