**Contemporary Mathematics**

Volume: 134;
1992;
377 pp;
Softcover

MSC: Primary 16; 17; 18; 81;

**Print ISBN: 978-0-8218-5141-8
Product Code: CONM/134**

List Price: $76.00

AMS Member Price: $60.80

MAA Member Price: $68.40

**Electronic ISBN: 978-0-8218-7725-8
Product Code: CONM/134.E**

List Price: $71.00

AMS Member Price: $56.80

MAA Member Price: $63.90

# Deformation Theory and Quantum Groups with Applications to Mathematical Physics

Share this page *Edited by *
*Jim Stasheff; Murray Gerstenhaber*

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.

#### Readership

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

# Table of Contents

## Deformation Theory and Quantum Groups with Applications to Mathematical Physics

- Contents v6 free
- Preface vii8 free
- Hopf algebra actions—revisited 110 free
- Link-diagrams, Yang Baxter equations, and quantum holonomy 1928
- Duality and topology of 3-manifolds 4554
- Algebras, bialgebras, quantum groups, and algebraic deformations 5160
- Generalized Moyal quantization on homogeneous symplectic spaces 93102
- A simple construction of bialgebra deformations 115124
- Integrable deformations of meromorphic equations on P1 (C) 119128
- Quantum groups with two parameters 129138
- Quantum group theoretic proof of the addition formula for continuous q-Legendre polynomials 139148
- q-special functions, a tutorial 141150
- q-special functions and their occurrence in quantum groups 143152
- Quantum flag and Schubert schemes 145154
- Homological perturbation theory, Hochschild homology, and formal groups 183192
- Tannaka-Krein theorem for quasi-Hopf algebras and other results 219228
- Simple smash products 233242
- Quantum group of links in a handlebody 235244
- Quantum Poisson SU (2) and quantum Poisson spheres 247256
- Deformation cohomology for bialgebras and quasi-bialgebras 259268
- Drinfel'd's quasi-Hopf algebras and beyond 297306
- Hopf algebra techniques applied to the quantum group Uq(s/(2)) 309318
- Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories 325334
- Elementary paradigms of quantum algebras 351360