Representation Theory of Infinite-Dimensional Lie Superalgebras and W-algebras and Their Mathematical Applications

在这个项目下，作者与麻省理工学院的Victor G.Kac教授共同研究了仿射李超代数和超共形代数(SCA)的结构和表示理论，取得了一些显著的成果，具体如下：1.给出了经典仿射超代数sl(m|n)和osp(m|n)^的玻色子和费米子的基本表示的显式构造。利用这种构造，我们得到了基本sl(m|n)-模的Weyl-Kac型、theta-函数型和拟粒子型的特征标公式，并推导出了基本osp(m|n)-模的特征标的模性质。2.我们发现仿射超代数sl(m|L)^的基本表示的特征标是用Appell椭圆函数写成的，并利用这些特征标推导出了它们的特征标的渐近性。但是…更重要的是，它们之间存在着很大的差异。对于超共形代数，不存在不变双线性形式，也不存在Weyl群，而Weyl群是研究仿射超代数表示的最重要工具。正因为如此，超共形代数的表示的研究一直是极其困难的，关于它的严格的数学结果也只有几个。我们成功地利用Drinfeld-Sokolov约化的量子化构造了W-代数，这使得我们能够详细地研究超共形代数的结构和表示。4.顶点代数在W-代数理论中占有重要的地位。但“雷蒙德类粒子”不能在顶点代数的框架内处理。为了处理它们，我们需要考虑“扭曲”的顶点代数。因此，我们进一步将W-代数理论推广到扭曲顶点代数，构造了“扭曲超共形代数”，并研究了它们的结构和表示。5.W-代数理论是从仿射超代数到超共形代数的函数。即W-代数不仅给出了超共形代数的构造，而且将仿射超代数的表示变换为超共形代数的表示。这在超共形代数的表示理论方面取得了显着的进展。作为应用，我们给出了超共形代数的行列式公式、特征标公式和自由场表示。较少

Under this project, the author has made joint research with Professor Victor G. Kac of Massachusetts Institute of Technology and obtained some remarkable results on the structure and representation theory of affine Lie superalgebras and supercoformal algebras (SCA), which are described as follows :1.We gave explicit construction of basic representations of classical affine superalgebras sl(m|n)^and osp(m|n)^in terms of bosons and fermions. Using this construction, we obtained character formulas of Weyl-Kac type and of theta-function type and of quasi-particle-type for basic sl(m|n)^modules and deduced modular properties of characters of basic osp(m|n)^-modules.2.We found that the characters of basic representations of affine superalgebras sl(m|l)^are written in terms of Appell's elliptic functions and, by using them, deduced the asymptotics of their characters.3.Most important classes among infinite-dimensional superalgebras are "affine superalgebrad" and "superconformal algebras". But … More there are quite differences between them. For superconformal algebras there exist no invariant bilinear forms nor Weyl groups, which are most important tools for the study of representations of affine superalgebras. Because of this reason, the study of representation of superconformal algebras has been extremely difficult and there were only a few rigorous mathematical results about it. We succeeded to construct W-algebras in terms of the quantization of Drinfeld-Sokolov reduction, which enabled us to study the structure and representations of superconformal algebras in detail.4.Vertex algebras play important roles in the theory of W-algebras. But "Ramond-type particles" cannot be treated in the framework of vertex algebras. In order to treat them, we need to consider "twisted" vertex algebras. So we went further to extend our theory of W-algebras to twisted vertex algebras and constructed "twisted superconformal algebras" and studied their structures and representations.5.The theory of W-algebras is the "function" from affine superalgebras to superconformal algebras. Namely W-algebras do not only give construction of superconformal algebras, but also transform representations of affine superalgebras to representations of superconformal algebras. This gave remarkable progress in the representation theory of superconformal algebras. As its application, we gave determinant formulas, character formulas and free fields representations of superconformal algebras. Less