Contemporary Mathematics 1992; 377 pp; softcover Volume: 134 ISBN10: 0821851411 ISBN13: 9780821851418 List Price: US$67 Member Price: US$53.60 Order Code: CONM/134
 Quantum groups are not groups at all, but special kinds of Hopf algebras of which the most important are closely related to Lie groups and play a central role in the statistical and wave mechanics of Baxter and Yang. Those occurring physically can be studied as essentially algebraic and closely related to the deformation theory of algebras (commutative, Lie, Hopf, and so on). One of the oldest forms of algebraic quantization amounts to the study of deformations of a commutative algebra \(A\) (of classical observables) to a noncommutative algebra \(A_h\) (of operators) with the infinitesimal deformation given by a Poisson bracket on the original algebra \(A\). This volume grew out of an AMSIMSSIAM Joint Summer Research Conference, held in June 1990 at the University of Massachusetts at Amherst. The conference brought together leading researchers in the several areas mentioned and in areas such as "\(q\) special functions", which have their origins in the last century but whose relevance to modern physics has only recently been understood. Among the advances taking place during the conference was Majid's reconstruction theorem for Drinfel'd's quasiHopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory. Readership Research mathematicians and graduate students and their counterparts in mathematical physics. Table of Contents  M. Cohen  Hopf algebra actionsrevisited
 P. CottaRamusino and M. Rinaldi  Linkdiagrams, Yang Baxter equations, and quantum holonomy
 L. Crane  Duality and topology of \(3\)manifolds
 M. Gerstenhaber and S. D. Schack  Algebras, bialgebras, quantum groups, and algebraic deformations
 J. M. GraciaBondía  Generalized Moyal quantization on homogeneous symplectic spaces
 R. Grossman and D. Radford  A simple construction of bialgebra deformations
 G. F. Helminck  Integrable deformations of meromorphic equations on \(\mathbb P^1(\mathbb C)\)
 N. H. Jing  Quantum groups with two parameters
 H. T. Koelink  Quantum group theoretic proof of the addition formula for continuous \(q\)Legendre polynomials
 H. T. Koelink and T. H. Koornwinder  \(q\)special functions, a tutorial
 T. H. Koornwinder  \(q\)special functions and their occurrence in quantum groups
 V. Lakshmibai and N. Reshetikhin  Quantum flag and Schubert schemes
 L. A. Lambe  Homological perturbation theory, Hochschild homology, and formal groups
 S. Majid  TannakaKrein theorem for quasiHopf algebras and other results
 S. Montgomery  Simple smash products
 J. H. Przytycki  Quantum group of links in a handlebody
 A. J. L. Sheu  Quantum Poisson \(SU(2)\) and quantum Poisson spheres
 S. Shnider  Deformation cohomology for bialgebras and quasibialgebras
 J. Stasheff  Drinfel'd's quasiHopf algebras and beyond
 M. Takeuchi  Hopf algebra techniques applied to the quantum group \(U_q(sl(2))\)
 D. N. Yetter  Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories
 C. Zachos  Elementary paradigms of quantum algebras
