(特征零域上)Cartan型李代数是最重要的无限维李代数之一,在微分几何和其他数学分支都有重要应用。近年来,李代数表示理论发展迅速,伴随着新思想的建立和新方法的发现,许多长期未解的经典公开问题得到解决,为该领域的发展提供了新的动力。以此为契机,Cartan型李代数的表示理论也有显著进展,但其结果比较零散,未能形成完整的理论体系,许多重要问题仍未解决,许多未知领域尚需开拓。本项目将依托课题组多年从事相关研究的良好基础,努力吸取前人的研究成果,对Cartan型李代数及相关李代数的表示理论进行系统深入的研究。我们将在包含下列课题的研究上做出创新性的研究成果:不可约Harish-Chandra模的分类问题,广义最高权模的结构与特征标公式,特定条件下模的分类与刻画,权空间无限维和非权单模的构造,Cartan型李代数与Cartan型模李(超)代数及其他李代数表示理论的联系等。

The Cartan type Lie algebras (over fields of characteristic 0), which have many applications in differential geometry and other mathematical branches, are one of the most important classes of infinite-dimensional Lie algebras. In recent years, with the discoveries of new ideas and developments of new methods in the field of Lie algebra theory, large progresses have been made for the representations of Cartan type Lie algebras. In particular, some classical problems, that had been open for a very long time, were settled successfully, attracting many attentions from Lie algebraists of various interests and providing motivations of new developments. However, the results for Cartan type Lie algebras are rather scattered, many important problems remain unsettled and many new research areas need to be explored. Based on the already established theories, known results gained by other mathematicians in this field, and the research work we have done all these years, the present project will make deep and systematical studies in the research of representation theory of Cartan type Lie algebras and other related algebras. We shall obtain original results in the subjects including: classification of irreducible Harish-Chandra modules, characterizations and realizations of generalized highest weight modules, classifications of special classes of modules under certain conditions, constructions of irreducible modules with infinite-dimensional weight spaces or non-weight modules, the relations between the representations of Cartan type Lie algebras and their counterparts over fields of positive characteristic and other related algebras.