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数学进展
研究论文
量子群理论概述
A Survey of the Theory of Quantum Groups

王建磐
Wang Jian-pan

华东师范大学
(Department of Mathematics, East China Normal University

收稿日期: 1991-12-25
出版日期: 1991-10-15

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摘要  量子群理论是近年来新兴的一个数学分支,它起源于理论物理,是由苏联数学物理学家(1990年Fields奖得主之一)及其合作者们在用“量子反散射方法”研究量子力学中的量子可积系统时最先提出来的.从某种意义上说,量子群的研究就是Hopf代数的研究,而Hopf代数与数学的许多分支都密切相关。因此,量子群理论一进入数学,很快引起了许多背景不同的数学家的兴趣,发展得相当迅速。例如,环论专家花更大的气力研究Hopf代数的结构,并且研究如何从交换的或余交换的Hopf代数经过某种变形理论得到既非交换亦非余交换的Hopf代数(量子化),或用某种特殊的方法(例如,苏联数
Abstract:The theory of quantum groups is a new branch of mathematics developed in the last few years, having its source in theoretical physics. The theory is progressing quite rapidly, appealing to many mathematicians with various backgrounds. The present paper gives a brief but comprehensive introduction to the mathematical theory of quantum groups, covering such important topics as the definitions of Hopf algebras and quantum groups, quantizations of functional algebras and (the enveloping algebras of) Lie algebras, the duality theory, the constructions of quantum groups via quadratic algebras and via R-matrices, the Yang-Baxter equation, Hecke algebras and q-Schur algebras, compact matrix pseudo-groups, as well the representation theory of quantum groups, both the viewpoint of the corepresentations of quantized functional algebras and the viewpoint of the representations of quantized enveloping algebras. The paper provides for the readers who are interested in the theory of quantum groups an express way to an overall understanding of the theory, so that they will be able to go further to study a certain aspect deeper and to make their own progress.
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