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Deformation Theory and Quantum Groups with Applications to Mathematical Physics
 
Deformation Theory and Quantum Groups with Applications to Mathematical Physics
eBook ISBN:  978-0-8218-7725-8
Product Code:  CONM/134.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Deformation Theory and Quantum Groups with Applications to Mathematical Physics
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Deformation Theory and Quantum Groups with Applications to Mathematical Physics
eBook ISBN:  978-0-8218-7725-8
Product Code:  CONM/134.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
  • Book Details
     
     
    Contemporary Mathematics
    Volume: 1341992; 377 pp
    MSC: 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 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
     
     
    • Articles
    • Miriam Cohen — Hopf algebra actions—revisited [ MR 1187276 ]
    • Paolo Cotta-Ramusino and Maurizio Rinaldi — Link-diagrams, Yang-Baxter 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. Gracia-Bondía — 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 group-theoretic 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 — Tannaka-Kreĭn theorem for quasi-Hopf algebras and other results [ MR 1187289 ]
    • Susan Montgomery — Simple Smash Products
    • Józef H. Przytycki — Quantum group of links in a handlebody [ MR 1187291 ]
    • Albert Jeu-Liang Sheu — Quantum Poisson ${\rm SU}(2)$ and quantum Poisson spheres [ MR 1187292 ]
    • Steven Shnider — Deformation cohomology for bialgebras and quasi-bialgebras [ MR 1187293 ]
    • Jim Stasheff — Drinfel′d’s quasi-Hopf 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 ]
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1341992; 377 pp
MSC: 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 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.

  • Articles
  • Miriam Cohen — Hopf algebra actions—revisited [ MR 1187276 ]
  • Paolo Cotta-Ramusino and Maurizio Rinaldi — Link-diagrams, Yang-Baxter 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. Gracia-Bondía — 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 group-theoretic 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 — Tannaka-Kreĭn theorem for quasi-Hopf algebras and other results [ MR 1187289 ]
  • Susan Montgomery — Simple Smash Products
  • Józef H. Przytycki — Quantum group of links in a handlebody [ MR 1187291 ]
  • Albert Jeu-Liang Sheu — Quantum Poisson ${\rm SU}(2)$ and quantum Poisson spheres [ MR 1187292 ]
  • Steven Shnider — Deformation cohomology for bialgebras and quasi-bialgebras [ MR 1187293 ]
  • Jim Stasheff — Drinfel′d’s quasi-Hopf 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 ]
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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