This volume presents research conducted between 1989 and 1991 by the participants in the Leningrad Seminar on representation theory, dynamical systems, and their applications, headed by A. M. Vershik. The primary areas covered here are mathematical physics, Lie groups and their representations, infinite-dimensional groups, topology, and dynamical systems. The book contains a number of useful introductory surveys; for example, one paper by Vaksman and Soibelman provides a systematic description of the theory of quantum groups in the spirit of representation theory - a new and popular area for which there are few introductory surveys. A portion of the book is devoted to adic transformations and substitutions, a new area of ergodic theory. With a balance of survey papers and frontier research results, this book will appeal to graduate students and researchers alike.
Part I. Quantum Groups, Representations of Groups and Algebras: On some problems in the theory of quantum groups by Ya. S. Soibelman and L. L. Vaksman Representations of quantum $*$-algebras $s1 t(N+1,\Bbb R)$ by E. Ye. Vaysleb Generalized Hall-Littlewood symmetric functions and orthogonal polynomials by S. V. Kerov Formulation of Bell type problems and ""noncommutative"" convex geometry by A. M. Vershik and B. S. Tsirelson Projective representations of the infinite symmetric group by M. L. Nazarov Structure theorems for a pair of unbounded selfadjoint operators satisfying a quadratic relation by V. L. Ostrovskii and Yu. S. Samoilenko Relations between compact quantum groups and Kac algebras by L. I. Vainerman Combinatorial complexity of orbits in representations of the symmetric group by A. I. Barvinok Part II. Dynamical Systems and Approximations: Adic models of ergodic transformations, spectral theory, substitutions, and related topics by A. M. Vershik and A. N. Livshits Periodic metrics by D. Yu. Burago Flows with positive entropy by D. Yu. Burago On the spectral theory of adic transformations by B. Solomyak On simultaneous action of Markov shift and adic transformation by M. Solomyak About a certain weakly mixing substitution by A. N. Livshits Manifolds with intrinsic metric, and nonholonomic spaces by V. N. Berestovskii and A. M. Vershik.