Deformation Theory and Quantum Groups with Applications to Mathematical Physics: Proceedings of a AMS-IMS-SIAM 1990 Joint Summer Research Conference Held June 14-20 at the University of Massachusetts, Amherst, with Support from the National Science Foundation

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American Mathematical Soc., 1992 - Mathematics - 377 pages
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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 Drinfeld's quasi-Hopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory.
 

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Contents

Hopf algebra actionsrevisited
ix
Linkdiagrams Yang Baxter equations and quantum holonomy
19
Duality and topology of 3manifolds
45
Algebras bialgebras quantum groups and algebraic deformations
51
Generalized Moyal quantization on homogeneous symplectic spaces
93
A simple construction of bialgebra deformations
115
Integrable deformations of meromorphic equations on P1 C
119
Quantum groups with two parameters
129
Homological perturbation theory Hochschild homology and formal groups
183
TannakaKrein theorem for quasiHopf algebras and other results
219
Simple smash products
233
Quantum group of links in a handlebody
235
Quantum Poisson SU 2 and quantum Poisson spheres
247
Deformation cohomology for bialgebras and quasibialgebras
259
Drinfelds quasiHopf algebras and beyond
297
Hopf algebra techniques applied to the quantum group Uqs2
309

Quantum group theoretic proof of the addition formula for continuous qLegendre polynomials
139
qspecial functions a tutorial
141
qspecial functions and their occurrence in quantum groups
143
Quantum flag and Schubert schemes
145
Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories
325
Elementary paradigms of quantum algebras
351
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