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Elliptic quantum groups. Representations and related geometry. (English) Zbl 1467.17001

SpringerBriefs in Mathematical Physics 37. Singapore: Springer (ISBN 978-981-15-7386-6/pbk; 978-981-15-7387-3/ebook). xiii, 131 p. (2020).
This book, based on several lectures delivered by the author since 1998, presents a survey on recent developments on elliptic quantum groups, their representations and related topics. The presentation is as well very recommendable as a brief introduction to the subject. Elliptic quantum groups are quantum groups associated with elliptic solutions of the Yang-Baxter equation, and they divide into two types: the vertex type and the face type, the last one connected to dynamical quantum groups. The exposition focuses mainly on the face type elliptic quantum group \(U_{q, p}(\widehat{\mathfrak{sl}_2})\) associated with the affine Lie algebra \(\widehat{\mathfrak{sl}_2}\).
The book starts with an introduction to the subject, discussing its genesis, related topics, and different formulations. Chapters 2 to 5 concern the study of \(U_{q, p}(\widehat{\mathfrak{sl}_2})\), with regard to its Hopf algebroid structure, its (dynamical) representations and related vertex operators. Based on the explicit realization of vertex operators, Chapter 6 presents a derivation of elliptic weight functions and studies their basic properties. Chapter 7 discusses a tensor product of evaluation representations and a connection with elliptic weight functions. Chapter 8 presents a relation between a trace composition of vertex operators and solutions of the face type elliptic \(q\)-KZ equation. A geometrical interpretation of the previous results is provided in Chapter 9. The book contains five appendices, most of them summarizing distinct definitions relevant to the subject.

MSC:

17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B38 Yang-Baxter equations and Rota-Baxter operators
20G42 Quantum groups (quantized function algebras) and their representations
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