李超代数中若干问题研究

项目名称： 李超代数中若干问题研究

项目编号： No.11471090

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 刘文德

作者单位： 哈尔滨师范大学

项目金额： 60万元

中文摘要： 李超代数与理论物理及数学的许多分支有密切关系,本项目旨在研究李超代数中若干互相关联的重要问题：（1）类比Kac的识别定理及Benkart-Gregory-Premet给出的加细证明，建立模李超代数的识别定理；（2）利用权导子方法，计算典型模李超代数的系数在Cartan型李超代数的上同调群；（3）类比模李代数方法，刻画Cartan型模李超代数无穷小形变；（4）利用Kac模研究奇Cartan型模李超代数的不可约表示；（5）刻画非限制Cartan型模李超代数的阶化极大子代数以及限制Cartan型李超代数与特征零Cartan型李超代数的非阶化极大子代数；（6）研究可解李超代数的上同调群的超代数结构以及忠实表示相关问题；（7）研究限制色李代数和限制Hom-李超代数的导子、上同调、形变等问题；（8）研究仿射Kac-Moody超代数，逆步李超代数的超导子代数的结构, 仿射李超代数生成元配对问题。

中文关键词： 李超代数；结构；表示；分类；上同调

英文摘要： The theory of Lie superalgebras is an important branch of Lie theory, which is closely related to Physics and many branches of Mathematics. We are mainly concerned with some key problems relative to the classification and representation theory of simple Lie superalgebras: (1) Following the Recognition Theorem due to Kac and the refined proof due to Benkart, Gregory and Premet, we aim to establish a corresponding super-version theorem; (2) Using the weight-derivation method, we compute the low dimensional cohomology groups of classical modular Lie superalgebras with coefficients in the modular Lie superalgebras of Cartan type;(3) As in modular Lie algebra case, we aim to determine the infinitesimal deformations of modular Lie superalgebras of Cartan type; (4) Using Kac modules, we aim to classify the irreducible representations of certain simple modular Lie superalgebras of odd Cartan type; (5) We aim to characterize the maximal graded subalgebras of the non-restricted graded simple modular Lie superalgebras of Cartan type and the nongraded maximal subalgebras of the restricted Lie superalgebras of Cartan type or the Cartan Lie superalgebras of characteristic zero; (6) We aim to study the superalgebra structure of cohomology and the problems on the faithful representations for some solvable Lie superalgebras over a field of any characteristic;(7) Using the theory of modular Lie superalgebras, we aim to research the derivations、cohomology and deformations of restricted Lie color algebras and restricted Hom-Lie superalgebras; (8) We aim to study affine Kac-moody superalgebras, characterize superderivation superalgebras of the contragredient Lie superalgebras and determine the pairings of generators of affine Lie superalgebras.

英文关键词： Lie superalgebra;structure;representation;classification;cohomology