无限维李代数的表示是当今李理论的中心课题之一，也是理论物理关注的热点。随着有限维单李代数，仿射李代数，Virasoro 李代数等等这些经典李代数的表示论研究的不断深入和成熟,高秩无限维李代数的表示逐渐引起了许多数学家和物理学家的关注。Witt 李代数，toroidal 李代数，全toroidal李代数，扩张仿射李代数(EALA)，等等都是重要的高秩无限维李代数。本项目申请人将依托近些年来积累的专业知识基础，努力吸取前研究人的研究成果，研究与量子环面相关的李代数的表示。具体地说研究量子环面的分次不可约模；量子环面的导子李代数(包含Witt代数)的不可约Harish-Chandra模的分类和构造；坐标代数为量子环面的全toroidal李代数以及扩张仿射李代数的权空间有限维的不可约可积模的分类和构造；以及这些李代数的权空间无限维的权模与非权模的构造。

The representation theory of infinite dimensional Lie algebra is an important part of the theory of Lie algebra, is also the focus of mathematical physics. Recently, developing representation theory for higher rank infinite dimensional Lie algebra has attracted extensively attention of many mathematicians and physicists. These Lie algebras include quantum tori Lie algebras, toroidal Lie algebras, full toroidal Lie algebras and extended affine Lie algebras, Witt algebras, derivation algebras of quantum tori and so on. In the current project, we discuss weight modules and nonweight modules over these Lie algebras related to quantum tori.