量子顶点代数理论和量子仿射代数

发布时间：2018-06-11 11:27

本文选题：量子顶点代数 + 量子仿射代数　；参考：《中国科学:数学》2017年11期

【摘要】：在广义的顶点代数领域中,一个基本的公开问题是,建立一个适当的量子顶点代数理论使得量子仿射代数和量子顶点代数自然地联系起来.部分地受Etingof和Kazhdan的量子顶点算子代数理论的启发,自2005年,作者系统地发展和研究了一个(弱)量子顶点代数及其拟模和φ-坐标拟模理论,建立了一些经典代数(如双杨氏代数)同量子顶点代数的自然联系,特别是最终给出了量子仿射代数同该意义下的弱量子顶点代数的一个自然联系.在此联系中,相对应的弱量子顶点代数在理论上存在,但其具体结构仍需要进一步去确定,并需证明它们是量子顶点代数.在某种程度上讲,这给所提的公开问题提供了一个初步答案.另一方面,这个理论在其发展的同时已被用来建立一些重要的代数同量子顶点代数的联系,显示了该理论的实用价值.本篇综述概括总结作者在这方面的主要结果,其中包括Zamolodchikov-Faddeev代数、无中心双杨氏代数、量子βγ-系统和量子仿射代数同(弱)量子顶点代数的联系.
[Abstract]:In the field of generalized vertex algebra, a basic open problem is to establish an appropriate quantum vertex algebra theory so that the quantum affine algebra and the quantum vertex algebra are connected naturally. Inspired in part by Etingof and Kazhdan's theory of quantum vertex operator algebra, since 2005, the author has systematically developed and studied a (weak) quantum vertex algebra and its quasi-module and 蠁 -coordinate pseudomodule theory. In this paper, we establish the natural relation between some classical algebras (such as double Young algebras) and quantum vertex algebras, especially, we give a natural relation between quantum affine algebras and weak quantum vertex algebras in this sense. In this connection, the corresponding weak quantum vertex algebras exist in theory, but their specific structures still need to be further determined, and they need to be proved to be quantum vertex algebras. To some extent, this provides a preliminary answer to the open question. On the other hand, this theory has been used to establish the relation between some important algebras and quantum vertex algebras, which shows the practical value of the theory. In this paper, the author's main results in this field are summarized, including Zamolodchikov-Faddeev algebra, non-centroid double Young algebra, quantum 尾纬 -system and quantum affine algebra and the relation between (weak) quantum vertex algebra and (weak) quantum vertex algebra.
【作者单位】： Department
【基金】：国家自然科学基金(批准号:11471268和11571391)资助项目
【分类号】：O152.5

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