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Quantum Groups in Three-Dimensional Integrability

  • Book
  • © 2022

Overview

Authors:
  1. Atsuo Kuniba

    1. Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Japan

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  • Presents quantized coordinate ring as a main player dual to what is usually meant by quantum group in physics literature
  • Illustrates quantization of the conventional Yang–Baxter and reflection equations, related to 3D integrability
  • Leads to matrix product formulas for R and K matrices having intriguing applications

Part of the book series: Theoretical and Mathematical Physics (TMP)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Introduction

    • Atsuo Kuniba
    Pages 1-8

  3. Tetrahedron Equation

    • Atsuo Kuniba
    Pages 9-19

  4. 3D R From Quantized Coordinate Ring of Type A

    • Atsuo Kuniba
    Pages 21-52

  5. 3D Reflection Equation and Quantized Reflection Equation

    • Atsuo Kuniba
    Pages 53-63

  6. 3D K From Quantized Coordinate Ring of Type C

    • Atsuo Kuniba
    Pages 65-89

  7. 3D K From Quantized Coordinate Ring of Type B

    • Atsuo Kuniba
    Pages 91-105

  8. Intertwiners for Quantized Coordinate Ring \(A_q(F_4)\)

    • Atsuo Kuniba
    Pages 107-116

  9. Intertwiner for Quantized Coordinate Ring \(A_q(G_2)\)

    • Atsuo Kuniba
    Pages 117-137

  10. Comments on Tetrahedron-Type Equation for Non-crystallographic Coxeter Groups

    • Atsuo Kuniba
    Pages 139-145

  11. Connection to PBW Bases of Nilpotent Subalgebra of \(U_q\)

    • Atsuo Kuniba
    Pages 147-175

  12. Trace Reductions of \(RLLL=LLLR\)

    • Atsuo Kuniba
    Pages 177-197

  13. Boundary Vector Reductions of \(RLLL=LLLR\)

    • Atsuo Kuniba
    Pages 199-212

  14. Trace Reductions of \(RRRR=RRRR\)

    • Atsuo Kuniba
    Pages 213-244

  15. Boundary Vector Reductions of \(RRRR=RRRR\)

    • Atsuo Kuniba
    Pages 245-255

  16. Trace Reduction of \((LGLG)K= K(GLGL)\)

    • Atsuo Kuniba
    Pages 257-272

  17. Boundary Vector Reductions of \((LGLG)K= K(GLGL)\)

    • Atsuo Kuniba
    Pages 273-283

  18. Reductions of Quantized \(G_2\) Reflection Equation

    • Atsuo Kuniba
    Pages 285-297

  19. Application to Multispecies TASEP

    • Prof. Atsuo Kuniba
    Pages 299-319

  20. Back Matter

    Pages 321-331
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Keywords

About this book

Quantum groups have been studied intensively in mathematics and have found many valuable applications in theoretical and mathematical physics since their discovery in the mid-1980s. Roughly speaking, there are two prototype examples of quantum groups, denoted by Uq and Aq. The former is a deformation of the universal enveloping algebra of a Kac–Moody Lie algebra, whereas the latter is a deformation of the coordinate ring of a Lie group. Although they are dual to each other in principle, most of the applications so far are based on Uq, and the main targets are solvable lattice models in 2-dimensions or quantum field theories in 1+1 dimensions.
 
This book aims to present a unique approach to 3-dimensional integrability based on Aq. It starts from the tetrahedron equation, a 3-dimensional analogue of the Yang–Baxter equation, and its solution due to work by Kapranov–Voevodsky (1994).
 
Then, it guides readers to its variety of generalizations, relations to quantum groups, and applications. They include a connection to the Poincaré–Birkhoff–Witt basis of a unipotent part of Uq, reductions to the solutions of the Yang–Baxter equation, reflection equation, G2 reflection equation, matrix product constructions of quantum R matrices and reflection K matrices, stationary measures of multi-species simple-exclusion processes, etc.
 
These contents of the book are quite distinct from conventional approaches and will stimulate and enrich the theories of quantum groups and integrable systems.



Authors and Affiliations

  • Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Japan

    Atsuo Kuniba

Bibliographic Information

  • Book Title: Quantum Groups in Three-Dimensional Integrability

  • Authors: Atsuo Kuniba

  • Series Title: Theoretical and Mathematical Physics

  • DOI: https://doi.org/10.1007/978-981-19-3262-5

  • Publisher: Springer Singapore

  • eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022

  • Hardcover ISBN: 978-981-19-3261-8Published: 26 September 2022

  • Softcover ISBN: 978-981-19-3264-9Published: 27 September 2023

  • eBook ISBN: 978-981-19-3262-5Published: 25 September 2022

  • Series ISSN: 1864-5879

  • Series E-ISSN: 1864-5887

  • Edition Number: 1

  • Number of Pages: XI, 331

  • Number of Illustrations: 64 b/w illustrations, 32 illustrations in colour

  • Topics: Mathematical Physics, Quantum Physics

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