Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations
Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations
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拟 Hopf 代数和 Knizhnik-Zamolodchikov 方程
DOI：
10.1007/978-3-642-84000-5_1
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发表时间：
1989
期刊：
影响因子：
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通讯作者：
V. Drinfeld
中科院分区：
文献类型：
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作者：
V. Drinfeld
本文是对文献[6]的简要论述。在§1中，我们提到了拟三角Hopf代数的概念，它是R-矩阵概念的抽象版本。在§2中，引入了拟三角拟Hopf代数的概念(用较弱的公理代替了余结合性)。在§3中，我们利用VG Knizhnik和AB Zamolodchikov提出的WZW理论中的n点函数的微分方程，构造了一类拟三角拟Hopf代数。定理1断言关于Planck常数的弱摄动理论实质上所有的拟三角Quazi-Hopf代数都属于这类。给出了两种辫子群表示等价的Kohno定理的一个自然证明。在§4中，我们讨论了纽结不变量的应用。在§5中，讨论了各种量子概念的经典极限。
This paper is a brief exposition of [6]. In § 1 we remind the notion of quasitriangular Hopf algebra which is an abstract version of the notion of R-matrix. In § 2 the notion of quasitriangular quasi-Hopf algebra is introduced (coassociativity is replaced by a weaker axiom). In § 3 we construct a class of quasitriangular quasi-Hopf algebras using the differential equations for n-point functions in the wzw theory introduced by vG Knizhnik and AB Zamolodchikov. Theorem 1 asserts that wi thin perturbation theory with respect to Planck's constant essentially all quasitriangular quazi-Hopf algebras belong to this class. A natural proof of Kohno's theorem on the equivalence of two kinds of braid group representations is given. In § 4 we discuss applications to knot invariants. In § 5 the classical limit of various quantum notions is discussed.