伪代数与无限维李代数结构与表示等相关问题的研究

One of the important researches in the modern algebra is to study pseudoalgebras and infinite dimensional Lie algebas. In this project, we study the structures and representations of some pseudoalgebras and infinite dimensional Lie algebras; the cohomology theory of pseudoalgebras and its applications in infinite dimensional Lie algebras or geometry. The main contents include the classifications of infinite generated simple Lie pseudoalgebras, and non-semisimple pseudo Leibniz (Lie) algebras and the left symmetric pseudoalgebras whose ranks are greater than and equal to 2; exsitences of the enveloping algebras for Lie pseudoalgebras; constructions and representations of some new Lie pseudoalgebras; the finite generated representations of high ranked Virasoro algebras, twisted Heisenberg-Virasoro algebras and Virasoro-Loop algebras in the category of modules over the enveloping algebras of their subalgebras; researches on categorification of pseudo sh algebras and the relations between 2-Lie pseudoalgebras and 2-category; the establishment of the (co)homology theory on the representation category of pseudoalgebras; researches on the relations between the automorphic group of pseudoalgebras and their subalgebras consisting of the fixed points, and their applications; the realizations of the geometric representations of Lie pseudoalgebras etc..This project further enriches and completes the structure and representation theory of pseudoalgebras and infinite dimensional Lie algebras, and provide some new research tools. It is a meaningful topic.

伪代数和无限维李代数的研究是当今代数学重要研究方向之一. 本项目主要研究某些伪代数和无限维李代数结构和表示；研究伪代数同调理论并探索伪代数在无限维李代数和几何中的应用.主要内容包括无限生成的单李伪代数和秩大于等于2的有限维非半单伪Leibniz（李）代数与左对称伪代数的分类；李伪代数的包络代数的存在性；某些新李伪代数的构造及其表示；高秩Virasoro代数， 扭Heisenberg-Virasoro代数和Virasoro-Loop代数在其子代数的包络代数表示范畴中有限生成表示；研究伪sh代数在高阶范畴中的实现并探索2-李伪代数和2-范畴之间的联系；试图在伪代数表示范畴上建立同调理论；研究伪代数自同构群与不动点子代数之间的关系及其应用；李伪代数几何表示的实现等. 本项目研究进一步丰富和完善了伪代数和无限维李代数结构和表示理论，并提供新的研究工具，是一个非常有意义的课题.

为了统一表示理论,Beilinson和Drinfeld提出了Chiral代数的概念。作为一类特殊的Chiral代数,伪代数近年来发展迅猛。本项目在B.Bakalov,A.D’Andrea和V.Kac等学者的工作基础上进一步围绕伪代数结构和分类,伪代数与无限维李代数表示,伪代数上同调及其应用等方面展开了研究并取得系列研究成果,如完整分类了低秩Leibniz共形代数以及某些低秩李伪代数;刻画了一系列与Virasoro共形代数相关的共形代数表示;证明了秩为二的李共形代数有限不可约表示都是秩为一的表示,并给出这些表示的构造;定义了一类新李伪代数W(g,\pi,b),并确定了其结构、有限单表示及上同调等;建立了结合伪代数中的Schur-Weyl对偶定理、Jacobson稠密性定理和Morita定理并给出它们的应用;建立了一类带有导子的李伪代数上同调理论,并由此建立了该类李伪代数的扩张及形变理论等。这些结果丰富了伪代数和无限维李代数结构理论、表示理论和同调理论,为进一步应用提供了理论基础。