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Complex Semisimple Quantum Groups and Representation Theory

  • Book
  • © 2020


  • Provides a comprehensive, accessible and self-contained introduction to the theory of quantized universal enveloping algebras and their associated quantized semisimple Lie groups
  • Presents complete proofs of many results that are otherwise scattered throughout the literature
  • Offers a unified approach to both the algebraic and the analytic theory of quantum groups using coherent conventions and notations
  • The first book to address the representation theory of general complex semisimple quantum groups

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2264)

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About this book

This book provides a thorough introduction to the theory of complex semisimple quantum groups, that is, Drinfeld doubles of q-deformations of compact semisimple Lie groups. The presentation is comprehensive, beginning with background information on Hopf algebras, and ending with the classification of admissible representations of the q-deformation of a complex semisimple Lie group.

 The main components are:

-   a thorough introduction to quantized universal enveloping algebras over general base fields and generic deformation parameters, including finite dimensional representation theory, the Poincaré-Birkhoff-Witt Theorem, the locally finite part, and the Harish-Chandra homomorphism,

-   the analytic theory of quantized complex semisimple Lie groups in terms of quantized algebras of functions and their duals,

-   algebraic representation theory in terms of category O, and

-   analytic representationtheory of quantized complex semisimple groups.

 Given its scope, the book will be a valuable resource for both graduate students and researchers in the area of quantum groups.

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Table of contents (6 chapters)


“The book is largely self-contained. … It is highly recommended for mathematicians of all levels wishing to learn about these topics, in the algebraic setting and/or in the analytic setting.” (Huafeng Zhang, zbMATH 1514.20006, 2023)

Authors and Affiliations

  • School of Mathematics & Statistics, University of Glasgow, Glasgow, UK

    Christian Voigt

  • Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, Aubière Cedex, France

    Robert Yuncken

About the authors

Christian Voigt is a Senior Lecturer at the School of Mathematics, University of Glasgow. His main research area is noncommutative geometry, with a focus on quantum groups, operator K-theory, and cyclic homology.

Robert Yuncken is Maître de Conférences at the Laboratoire de Mathématiques Blaise Pascal, Univerité Clermont Auvergne in France.  His main research interests are in operator algebras, geometry, and representation theory.

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