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Title: Introduction to quantized LIE groups and algebras

Abstract

In this paper, the authors give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups the authors study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then the authors explain in detail the concept of quantization for them. As an example the quantization of sl[sub 2] is explicitly carried out. Next, the authors show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction, the authors explicitly construct the universal R matrix for the quantum sl[sub 2] algebra. In the last section, the authors deduce all finite-dimensional irreducible representations for q a root of unity. The authors also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

Authors:
 [1]
  1. Inst. voor Theoretische Fysica, Valckenierstraat 65, 1018 XE Amsterdam (Netherlands)
Publication Date:
OSTI Identifier:
7135372
Resource Type:
Journal Article
Journal Name:
International Journal of Modern Physics A; (United States)
Additional Journal Information:
Journal Volume: 7:25; Journal ID: ISSN 0217-751X
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; QUANTIZATION; LIE GROUPS; CONFORMAL INVARIANCE; FIELD THEORIES; IRREDUCIBLE REPRESENTATIONS; POISSON EQUATION; QUANTUM MECHANICS; R MATRIX; TENSORS; USES; DIFFERENTIAL EQUATIONS; EQUATIONS; INVARIANCE PRINCIPLES; MATHEMATICS; MATRICES; MECHANICS; PARTIAL DIFFERENTIAL EQUATIONS; SYMMETRY GROUPS; 661100* - Classical & Quantum Mechanics- (1992-)

Citation Formats

Tjin, T. Introduction to quantized LIE groups and algebras. United States: N. p., 1992. Web. doi:10.1142/S0217751X92002805.
Tjin, T. Introduction to quantized LIE groups and algebras. United States. doi:10.1142/S0217751X92002805.
Tjin, T. Sat . "Introduction to quantized LIE groups and algebras". United States. doi:10.1142/S0217751X92002805.
@article{osti_7135372,
title = {Introduction to quantized LIE groups and algebras},
author = {Tjin, T},
abstractNote = {In this paper, the authors give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups the authors study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then the authors explain in detail the concept of quantization for them. As an example the quantization of sl[sub 2] is explicitly carried out. Next, the authors show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction, the authors explicitly construct the universal R matrix for the quantum sl[sub 2] algebra. In the last section, the authors deduce all finite-dimensional irreducible representations for q a root of unity. The authors also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.},
doi = {10.1142/S0217751X92002805},
journal = {International Journal of Modern Physics A; (United States)},
issn = {0217-751X},
number = ,
volume = 7:25,
place = {United States},
year = {1992},
month = {10}
}