
Elliptic Quantum Groups
Representations and Related Geometry
Hitoshi Konno(Author)
Springer (Publisher)
Published on 15. September 2020
Book
Paperback/Softback
XIII, 131 pages
978-981-15-7386-6 (ISBN)
Description
This is the first book on elliptic quantum groups, i.e., quantum groups associated to elliptic solutions of the Yang-Baxter equation. Based on research by the author and his collaborators, the book presents a comprehensive survey on the subject including a brief history of formulations and applications, a detailed formulation of the elliptic quantum group in the Drinfeld realization, explicit construction of both finite and infinite-dimensional representations, and a construction of the vertex operators as intertwining operators of these representations. The vertex operators are important objects in representation theory of quantum groups. In this book, they are used to derive the elliptic q-KZ equations and their elliptic hypergeometric integral solutions. In particular, the so-called elliptic weight functions appear in such solutions. The author's recent study showed that these elliptic weight functions are identified with Okounkov's elliptic stable envelopes for certain equivariant elliptic cohomology and play an important role to construct geometric representations of elliptic quantum groups. Okounkov's geometric approach to quantum integrable systems is a rapidly growing topic in mathematical physics related to the Bethe ansatz, the Alday-Gaiotto-Tachikawa correspondence between 4D SUSY gauge theories and the CFT's, and the Nekrasov-Shatashvili correspondences between quantum integrable systems and quantum cohomology. To invite the reader to such topics is one of the aims of this book.
Reviews / Votes
"The discussions in this book are brief and technical, so it will probably be of most interest to readers who are already familiar with other types of quantum groups but would like to get some idea of what is new or different in elliptic quantum groups." (Robert Harold McRae, Mathematical Reviews, April, 2022)"The presentation is as well very recommendable as a brief introduction to the subject." (Sonia Natale, zbMATH 1467.17001, 2021)
"The discussions in this book are brief and technical, so it will probably be of most interest to readers who are already familiar with other types of quantum groups but would like to get some idea of what is new or different in elliptic quantum groups." (Robert Harold McRae, Mathematical Reviews, April, 2022)
"The presentation is as well very recommendable as a brief introduction to the subject." (Sonia Natale, zbMATH 1467.17001, 2021)
More details
Product info
Book
Series
Edition
1st ed. 2020
Language
English
Place of publication
Singapore
Singapore
Target group
Professional and scholarly
Illustrations
3 s/w Abbildungen
XIII, 131 p. 3 illus.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 9 mm
Weight
236 gr
ISBN-13
978-981-15-7386-6 (9789811573866)
DOI
10.1007/978-981-15-7387-3
Schweitzer Classification
Other editions
Additional editions

E-Book
09/2020
Springer
€69.54
Available for download
Content
Preface.- Acknowledgements.- Chapter 1: Introduction.- Chapter 2: Elliptic Quantum Group.- Chapter 3: The H-Hopf Algebroid Structure of.- Chapter 4: Representations of.- Chapter 5: The Vertex Operators.- Chapter 6: Elliptic Weight Functions.- Chapter 7: Tensor Product Representation.- Chapter 8: Elliptic q-KZ Equation.- Chapter 9: Related Geometry.- Appendix A.- Appendix B.- Appendix C.- Appendix D.- Appendix E.- References.