两类无限李共形代数的上同调

由V.G. Kac引入的李共形代数在量子场论和顶点算子代数的理论中发挥了重要的作用。无限李共形代数，因与某些重要的非有限分次李代数的密切关系，得到了越来越多的研究。 . 一般李共形代数gc_N可能是最重要的一类无限李共形代数，这个代数所起的作用类似于一般线性李代数gl_N在李代数理论中的作用。V.G. Kac与他的合作者们在文献[BKV]中建立了李共形代数的一般上同调理论，并且提出了一个公开问题，就是计算gc_N的上同调群。Y. Su计算了gc_N的一些低维上同调群。本项目将致力于对此问题的研究，并力图对某些特殊的N值，彻底解决此问题。. 申请人的博士论文介绍了另外一类称为map Virasoro李共形代数的无限李共形代数。在已有的研究工作的基础上，本项目将继续对map Virasoro李共形代数进行研究，并计算它的一些低维上同调群。这是我们研究的另外一个目标。

Lie conformal algebras，firstly introduced by V. G. Kac，play important roles in quantuam field theory and vertex operator algebras. Because of their close relations with some important non-finitely Lie algebras, infinite Lie conformal algebras have drawn more and more attention in the literature...The most important example of infinite Lie conformal algebras may be the general Lie conformal algebra gc_N，which plays the same important role in the theory of Lie conformal algebras as the general linear algebras gl_N dose in the theory of Lie algebras. V.G. Kac and his collaborators develop the general cohomology theory of Lie conformal algebras and pose an open problem of computing the cohomology groups of gc_N in [BKV]. Y. Su compute some low dimensional cohomology groups of gc_N. In this project, we will mainly focus on this problem and try to solve it for some special values of N.. .Another class of infinite Lie conformal Lie algebras, called map Virasoro Lie conformal algebras, was introduced in the applicant's PH.D thesis. On the basis of our previous research works, we continue the study of map Virasoro Lie conformal algebras in this project and compute some low dimensional cohomology groups of map Virasoro Lie conformal algebras. This is another goal of our research.