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Deformation Theory and Quantum Groups with Applications to Mathematical Physics
Edited by: Murray Gerstenhaber and Jim Stasheff
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Contemporary Mathematics
1992; 377 pp; softcover
Volume: 134
ISBN-10: 0-8218-5141-1
ISBN-13: 978-0-8218-5141-8
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 AMS-IMS-SIAM 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 quasi-Hopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory.

Research mathematicians and graduate students and their counterparts in mathematical physics.

• M. Cohen -- Hopf algebra actions--revisited
• P. Cotta-Ramusino and M. Rinaldi -- Link-diagrams, 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. Gracia-Bondía -- 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 -- Tannaka-Krein theorem for quasi-Hopf 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 quasi-bialgebras
• J. Stasheff -- Drinfel'd's quasi-Hopf 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