Spinor and Oscillator Representations of Quantum Groups-猫眼课题宝

DOI：

10.1007/978-1-4612-0261-5_5

发表时间：

1994

期刊：

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其他

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作者：

Jintai Ding;I. Frenkel

量子群理论起源于统计力学和量子场论的完全可积模型(见[FRT]，[J4])。随着Drinfeld[DL]和Jim[JI]独立发现任意Kac-Moody代数的万能包络代数的q-变形，它成为一个真正的数学理论。这一显着的结果立即提出了许多关于与Kac-Moody代数有关的各种结构的Q-变形的问题。Lusztig[L]在这个方向上迈出了重要的一步，他得到了Kac-Moody代数关于一般参数或形式参数q的最高权表示范畴的Q-变形。在Lusztig的工作之前和之后，关于有限维仿射李代数[D2]，[J2]，[FJ]，[H]，[FR]等的各种数学结构的Q-变形都有了一些成功的结果。然而，每个特定的问题都需要自己的洞察力和运气相结合，在某些情况下还会出现巨大的困难。现有理论的主要复杂性是缺乏与有限维李代数和仿射李代数相关的两个最有趣的量子群的子类的一般不变定义。唯一可用的使用生成元和关系的通用定义模糊了量子群的不变代数和几何性质，并阻碍了一些重要的应用。在[FRT]中强调了量子群不变方法的必要性，其中给出了经典李代数的矩阵实现的Q-模拟。但即使是后一种定义也是基于杨-巴克斯特方程的一个显著但相当特殊的解，可以被视为最好的折衷方案。在这篇文章中，我们想要发展一种不变的方法来构造与量子群有关的各种表示理论。更具体地说，我们利用量子群的一般表示理论定义了量子Clifford代数和Weyl代数。我们证明了量子Clifford和Weyl代数的显式公式与

The theory of quantum groups originates in completely integrable models of statistical mechanics and quantum field theory (see [FRT],[J4]). It becomes a truly mathematical theory with the independent discovery by Drinfeld [Dl] and Jimbo [JI] of a q-deformation of the universal enveloping algebra of an arbitrary Kac-Moody algebra. This remarkable result immediately raised numerous questions about q-deformations of various structures associated to Kac-Moody algebras. A major step in this direction was done by Lusztig [L], who obtained a q-deformation of the category of highest weight representations of Kac-Moody algebras for generic or formal parameter q. Before and after the work of Lusztig, there were a number of successful results on q-deformation of various mathematical structures of finite dimensional and affine Lie algebras [D2],[J2],[FJ],[H],[FR], etc. However each particular problem required its own insight combined with good luck and in some cases presented formidable difficulties. The main complication with the existing theory is the absence of the general invariant definition of the two most interesting subclasses of quantum groups associated to finitedimensional and affine Lie algebras. The only available general definition using generators and relations obscures the invariant algebraic and geometric nature of quantum groups and impedes some important applications. The necessity of an invariant approach to quantum groups was stressed in [FRT], where a q-analogue of the matrix realization of classical Lie algebras was given. But even the latter definition is based on a remarkable but rather ad hoc solution of Yang-Baxter equation and can be viewed as the best available compromise.In this paper, we would like to develop an invariant approach to various representation theoretical constructions related to quantum groups. More specifically, we define quantum Clifford and Weyl algebras using the general representation theory of quantum groups. We show that the explicit formulas for quantum Clifford and Weyl algebras match the ones

Spinor and Oscillator Representations of Quantum Groups

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