超Virasoro型李超代数的相关研究

Infinite dimensional Lie superalgebras are important objects of study in the theory of Lie superalgebra because of their wide applications in many fields such as superconformal field theory and integrable system. Super Virasoro algebras are ones of the most important infinite dimensional Lie superalgebras and can be obtained as affinizations of the only simple Balinsky-Novikov(BN) superalgebra in the complex number field. This project will study the so-called super Virasoro type Lie superalgebras which can be obtained as the affinizations of BN superalgebras. The project mainly focuses on the following aspects..(1) We need study the classification of BN superalgebras, particularly for low dimension, and their properties such as the derivations, automorphisms and deformations..(2) We further study how the above properties manage the related properties of their corresponding super Virasoro type Lie superalgebras. In fact, this turns infinite dimensional problem into finite dimensional problem..(3) Some particular super Virasoro type Lie superalgebras will be studied thoroughly, such as the (universal) central extensions and properties of super Virasoro type Lie superalgebras with even part being W(2,2) Lie algebra. Also, the structure and related representaion of superalgebras of W(2,2) Lie algebra will be determined according to the classification of intermediates series modules of W(2,2) Lie algebra.

由于在超共形场论和可积系统等领域的应用，无限维李超代数已成为李超代数研究领域中的一个重要课题。超Virasoro代数是其中非常重要的代表，它可由复数域内唯一的单Balinsky-Novikov(BN)超代数仿射构造得到。该项目主要研究的是和超Virasoro代数类似可以由BN超代数仿射构造得到的无限维李超代数:超Virasoro型李超代数。研究内容主要包括以下几个方面：作为基础我们需要研究BN超代数的低维分类，它的导子、自同构和形变等性质；研究BN超代数的上述性质对其所对应的超Virasoro型李超代数相关性质的可控性，这实际上是把无限维的问题转化为有限维的问题来考虑。此外，对一些特别的超Virasoro型李超代数进行深入的研究，如研究偶部是W(2,2)李代数的超Virasoro型李超代数的(泛)中心扩张及性质，并根据W(2,2)李代数中间序列模的分类确定其超代数的结构及相关表示等。

本项目主要研究了三个方面的内容：（1）对可以由Balinsky-Novikov超代数仿射构造得到的无限维李超代数,即超Virasoro型李超代数进行了相关研究；（2）对生成指标为2的有限维左对称代数进行了研究；（3）对齐性空间上的测地线、爱因斯坦度量的相关研究。