## 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 FoundationQuantum 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

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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### Common terms and phrases

A-bimodule A-module algebra structure antipode associated automorphism bar construction bialgebra bialgebra deformation braided C*-algebras classical coadjoint coalgebra coboundary cochain complex cocycle coefficients cohomology commutative comodule comp algebra comultiplication consider coproduct corresponding counit Curtright decomposition defined definition deformation quantization deformation theory denote diagram differential Drinfeld dual element equivalent finite dimensional formal formula functions functor G-algebra Gerstenhaber given graded Hence Hochschild Hochschild cohomology holomorphic holonomy homology homotopy Hopf algebra identity integral invariant inverse irreducible isomorphism Koszul Kulish label Lemma Lie algebra Lie group linear link-diagrams Math matrix module monoidal category monomials Moyal multiplication nilpotent notation obtain operator perturbation Phys polynomials power series preprint proof Proposition quantum groups quantum holonomy quasi-Hopf algebra quasitriangular relations representation Reshetikhin resolution result ring satisfy semisimple spectral sequence standard monomials subalgebra symmetric symplectic tensor product Theorem transference problem twisted universal enveloping algebra vector space Yang-Baxter equation Zachos