Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2020 | Book

Introduction Read chapter Read first chapter

Complex Semisimple Quantum Groups and Representation Theory

Authors: Prof. Christian Voigt, Ph.D. Robert Yuncken

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

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 representation theory 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.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
The theory of quantum groups, originating from the study of integrable systems, has seen a rapid development from the mid 1980s with far-reaching connections to various branches of mathematics, including knot theory, representation theory and operator algebras, see (Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) (Amer. Math. Soc., Providence, RI, 1987), pp. 798–820; Lusztig, Introduction to Quantum Groups (Birkhäuser/Springer, New York, 2010); Chari and Pressley, A Guide to Quantum Groups (Cambridge University Press, Cambridge, 1995)). While the term quantum group itself has no precise definition, it is used to denote a number of related constructions, including in particular quantized universal enveloping algebras of semisimple Lie algebras and, dually, deformations of the algebras of polynomial functions on the corresponding semisimple groups. In operator algebras, the theory of locally compact quantum groups (Kustermans and Vaes, Ann. Sci. École Norm. Sup. (4) 33(6):837–934, 2000) is a powerful framework which allows one to extend Pontrjagin duality to a fully noncommutative setting.
Christian Voigt, Robert Yuncken
Chapter 2. Multiplier Hopf Algebras
Abstract
In this chapter we collect definitions and basic results regarding Hopf algebras and multiplier Hopf algebras. Algebras are not assumed to have identities in general. Throughout we shall work over an arbitrary base field \( \mathbb {K} \), and all tensor products are over \( \mathbb {K}\).
Christian Voigt, Robert Yuncken
Chapter 3. Quantized Universal Enveloping Algebras

In this chapter we collect background material on quantized universal enveloping algebras. We give in particular a detailed account of the construction of the braid group action and PBW-bases, and discuss the finite dimensional representation theory in the setting that the base field \( \mathbb {K} \) is an arbitrary field and the deformation parameter \( q \in \mathbb {K}^\times \) is not a root of unity. Our presentation mainly follows the textbooks (Jantzen, Lectures on Quantum Groups, vol. 6 of Graduate Studies in Mathematics; Joseph, Quantum Groups and Their Primitive Ideals, vol. 29 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]) and (Lusztig, Introduction to Quantum Groups, Modern Birkhäuser Classics).

Christian Voigt, Robert Yuncken
Chapter 4. Complex Semisimple Quantum Groups
Abstract
In this chapter we introduce our main object of study, namely complex semisimple quantum groups. We complement the discussion with some background material on locally compact quantum groups in general, and on compact quantum groups arising from q-deformations in particular.
Christian Voigt, Robert Yuncken
Chapter 5. Category
Abstract
In this chapter we study some aspects of the representation theory of quantized universal enveloping algebras with applications to the theory of Yetter-Drinfeld modules and complex quantum groups. Our main goal is a proof of the Verma module annihilator Theorem, following the work of Joseph and Letzter. We refer to (Farkas and Letzter, Quantized representation theory following Joseph, in Studies in Lie Theory, vol. 243 of Progr. Math.) for a survey of the ideas involved in the proof and background.
Christian Voigt, Robert Yuncken
Chapter 6. Representation Theory of Complex Semisimple Quantum Groups
Abstract
In this chapter, we discuss the representation theory of complex semisimple quantum groups. The appropriate notion of a G q-representation here is that of a Harish-Chandra module for G q, which means an essential \(\mathfrak {D}(G_q)\)-module with K q-types of finite multiplicity, see Sect. 6.2. In particular, the irreducible unitary representations of G q belong to this class.
Christian Voigt, Robert Yuncken
Backmatter
Metadata
Title
Complex Semisimple Quantum Groups and Representation Theory
Authors
Prof. Christian Voigt
Ph.D. Robert Yuncken
Copyright Year
2020
Electronic ISBN
978-3-030-52463-0
Print ISBN
978-3-030-52462-3
DOI
https://doi.org/10.1007/978-3-030-52463-0

Premium Partner

    Image Credits