Elliptic quantum groups : representations and related geometry /

Elliptic quantum groups : representations and related geometry /

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Bibliographic Details
Author / Creator:	Konno, Hitoshi, author.
Imprint:	Singapore : Springer, [2020]
Description:	1 online resource.
Language:	English
Series:	SpringerBriefs in Mathematical Physics ; volume 37 SpringerBriefs in mathematical physics ; v. 37.
Subject:	Quantum groups. Elliptic functions. Quantum groups. Elliptic functions. Algebra. Group theory. Mathematical physics. Ordered algebraic structures. Electronic books.
Format:	E-Resource Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/12607602

Hidden Bibliographic Details
ISBN:	9789811573873 9811573875 9811573867 9789811573866
Notes:	Includes bibliographical references. Description based on online resource; title from digital title page (viewed on November 05, 2020).
Summary:	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 authors recent study showed that these elliptic weight functions are identified with Okounkovs elliptic stable envelopes for certain equivariant elliptic cohomology and play an important role to construct geometric representations of elliptic quantum groups. Okounkovs 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 CFTs, 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.
Other form:	Print version: 9811573867 9789811573866
Standard no.:	10.1007/978-981-15-7 10.1007/978-981-15-7387-3

Description
Summary:	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 stableenvelopes 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.
Physical Description:	1 online resource.
Bibliography:	Includes bibliographical references.
ISBN:	9789811573873 9811573875 9811573867 9789811573866