eBook ISBN:  9780821877258 
Product Code:  CONM/134.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821877258 
Product Code:  CONM/134.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 134; 1992; 377 ppMSC: Primary 16; 17; 18; 81
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 quasiHopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory.
ReadershipResearch mathematicians and graduate students and their counterparts in mathematical physics.

Table of Contents

Articles

Miriam Cohen — Hopf algebra actions—revisited [ MR 1187276 ]

Paolo CottaRamusino and Maurizio Rinaldi — Linkdiagrams, YangBaxter equations and quantum holonomy [ MR 1187277 ]

Louis Crane — Duality and topology of $3$manifolds [ MR 1187278 ]

Murray Gerstenhaber and Samuel D. Schack — Algebras, bialgebras, quantum groups, and algebraic deformations [ MR 1187279 ]

José M. GraciaBondía — Generalized Moyal quantization on homogeneous symplectic spaces [ MR 1187280 ]

Robert Grossman and David Radford — A simple construction of bialgebra deformations [ MR 1187281 ]

G. F. Helminck — Integrable deformations of meromorphic equations on ${\bf P}^1({\bf C})$ [ MR 1187282 ]

N. H. Jing — Quantum groups with two parameters [ MR 1187283 ]

H. T. Koelink — Quantum grouptheoretic proof of the addition formula for continuous $q$Legendre polynomials [ MR 1187284 ]

H. T. Koelink and T. H. Koornwinder — $q$special functions, a tutorial [ MR 1187285 ]

T. H. Koornwinder — $q$special functions and their occurrence in quantum groups [ MR 1187286 ]

V. Lakshmibai and N. Reshetikhin — Quantum flag and Schubert schemes [ MR 1187287 ]

Larry A. Lambe — Homological perturbation theory, Hochschild homology, and formal groups [ MR 1187288 ]

Shahn Majid — TannakaKreĭn theorem for quasiHopf algebras and other results [ MR 1187289 ]

Susan Montgomery — Simple Smash Products

Józef H. Przytycki — Quantum group of links in a handlebody [ MR 1187291 ]

Albert JeuLiang Sheu — Quantum Poisson ${\rm SU}(2)$ and quantum Poisson spheres [ MR 1187292 ]

Steven Shnider — Deformation cohomology for bialgebras and quasibialgebras [ MR 1187293 ]

Jim Stasheff — Drinfel′d’s quasiHopf algebras and beyond [ MR 1187294 ]

Mitsuhiro Takeuchi — Hopf algebra techniques applied to the quantum group $U_q({\rm sl}(2))$ [ MR 1187295 ]

David N. Yetter — Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories [ MR 1187296 ]

Cosmas Zachos — Elementary paradigms of quantum algebras [ MR 1187297 ]


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
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 quasiHopf 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.

Articles

Miriam Cohen — Hopf algebra actions—revisited [ MR 1187276 ]

Paolo CottaRamusino and Maurizio Rinaldi — Linkdiagrams, YangBaxter equations and quantum holonomy [ MR 1187277 ]

Louis Crane — Duality and topology of $3$manifolds [ MR 1187278 ]

Murray Gerstenhaber and Samuel D. Schack — Algebras, bialgebras, quantum groups, and algebraic deformations [ MR 1187279 ]

José M. GraciaBondía — Generalized Moyal quantization on homogeneous symplectic spaces [ MR 1187280 ]

Robert Grossman and David Radford — A simple construction of bialgebra deformations [ MR 1187281 ]

G. F. Helminck — Integrable deformations of meromorphic equations on ${\bf P}^1({\bf C})$ [ MR 1187282 ]

N. H. Jing — Quantum groups with two parameters [ MR 1187283 ]

H. T. Koelink — Quantum grouptheoretic proof of the addition formula for continuous $q$Legendre polynomials [ MR 1187284 ]

H. T. Koelink and T. H. Koornwinder — $q$special functions, a tutorial [ MR 1187285 ]

T. H. Koornwinder — $q$special functions and their occurrence in quantum groups [ MR 1187286 ]

V. Lakshmibai and N. Reshetikhin — Quantum flag and Schubert schemes [ MR 1187287 ]

Larry A. Lambe — Homological perturbation theory, Hochschild homology, and formal groups [ MR 1187288 ]

Shahn Majid — TannakaKreĭn theorem for quasiHopf algebras and other results [ MR 1187289 ]

Susan Montgomery — Simple Smash Products

Józef H. Przytycki — Quantum group of links in a handlebody [ MR 1187291 ]

Albert JeuLiang Sheu — Quantum Poisson ${\rm SU}(2)$ and quantum Poisson spheres [ MR 1187292 ]

Steven Shnider — Deformation cohomology for bialgebras and quasibialgebras [ MR 1187293 ]

Jim Stasheff — Drinfel′d’s quasiHopf algebras and beyond [ MR 1187294 ]

Mitsuhiro Takeuchi — Hopf algebra techniques applied to the quantum group $U_q({\rm sl}(2))$ [ MR 1187295 ]

David N. Yetter — Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories [ MR 1187296 ]

Cosmas Zachos — Elementary paradigms of quantum algebras [ MR 1187297 ]