Infinite-Dimensional Lie (Super) Algebras and Related Algebraic Structures

9970007这个建议的主要部分涉及代数结构，称为共形代数，这是共形场论中手征场的算子乘积展开的公理化描述。在共形代数的结构理论中使用的主要工具是解析伪群的Cartan-Lie理论。 因此，为了发展共形超代数的结构理论，需要将Cartan-Lie理论扩展到超情形。 简单解析超级伪群的分类出乎意料地变得非常丰富和美丽。 这些新物体需要仔细研究。 多维共形代数的结构理论的工作也在进行中。 该计划的其他内容包括W_infinity型李代数的表示理论、仿射李超代数的特征标理论及其与数论的联系、量子轨道理论等，我们有充分的理由相信，新的简单超赝群理论将在进一步理解标准模型、电磁场、强相互作用和弱相互作用的量子理论等方面发挥基础性作用。 共形代数的理论与无限维李代数和超代数的理论有着密切的联系，并可能导致更好地理解现实量子场论中的算子乘积展开。 仿射李超代数的特征标理论与模形式理论的一些最新发展密切相关。

9970007The main part of this proposal deals with the algebraic structure, called a conformal algebra, which is an axiomatic description of the operator product expansion of chiral fields in conformal field theory. The main tool used in the structure theory of conformal algebras is the Cartan-Lie theory of analytic pseudogroups. Thus, in order to develop a structure theory of conformal superalgebras one needs an extension of the Cartan-Lie theory to the super case. The classification of simple analytic super pseudogroups quite unexpectedly turned out to be very rich and beautiful. These new objects require a careful study. The work is also underway on a structure theory of multidimensional conformal algebras. Other items of the proposal include representation theory of W_infinity type of Lie algebras, a character theory for affine Lie superalgebras and its connections to number theory, and the theory of quantum orbifolds.There are deep reasons to believe that the new theory of simple super pseudogroups should play a fundamental role in further understanding of the Standard model, the quantum theory of electromagnetic, strong and weak interactions. The theory of conformal algebras is deeply related to the theory of infinite-dimensinal Lie algebras and superalgebras and may lead to a better understanding of the operator product expansion in realistic quantum field theories. The character theory of affine Lie superalgebras is intimately connected to some recent developments in the theory of modular forms.