Kac-Moody 代数及相关李代数的表示理论

Kac-Moody Lie algebras, which have many applications in theoretical physics and other mathematical branches, are among the most important classes of Lie algebras. The representation theory of the finite-type and affine-type algebras has been investigated extensively and deeply; however, there are still many important classical problems that remain unsettled and new areas that need to be developed; in particular, the representation theory of the indefinite-type algebras have not yet been studied systematically. In recent years, with the discovery of new methods and establishment of new theories in the field of Lie algebras, large progresses have been made for the representation theory of Kac-Moody algebras. Especially, some important problems, that had been open for a very long time, were settled successfully; this attracted many attentions from Lie algebraists with various interests and provided motivations for further developments. Based on the already established theories, known results developed by other mathematicians in this field, and the research work we have done all these years, the present project will make deeper investigations in the research of representation theory of Kac-Moody algebras and other related algebras. We shall obtain original results in the areas including: study of tensor product modules of various Harish-Chandra modules, classifications and characterizations of irreducible modules under certain nice conditions, constructions of irreducible modules with infinite-dimensional weight spaces, constructions and studies of Whittaker modules, constructions of other irreducible non-weight modules for affine-type Kac-Moody algebras; a first attempt to the study of the representation theory of indefinite-type Kac-Moody algebras; the relation among the representations of Kac-Moody algebras and other related algebras, such as Virasoro algebra, Witt algebras, EALA algebras, toroidal algebras, Cartan-type algebras, VOAs, quantum groups and so on.

Kac-Moody代数是最重要的李代数之一，在理论物理和其他数学分支都有重要应用。有限型和仿射型代数的结构与表示理论已相当丰富，但仍有许多重要的经典问题需要解决，以及许多未知领域需要开拓;不定型代数的表示理论尚未形成系统。近年来，随着李代数表示理论新思想的引进和新方法的建立，Kac-Moody 代数的表示理论也进一步丰富，尤其是一些重要问题的解决对该领域的发展提供了新的动力；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，继续深入研究Kac-Moody 代数及相关李代数的表示理论。我们将在包含下列课题的研究上做出创新性的研究成果：仿射李代数Harish-Chandra 模的张量积的研究，特定条件下模的分类与刻画，权空间无限维的模的构造，Whittaker 模的构造与研究，其他非权模的构造，不定型李代数表示理论的探索,Kac-Moody 代数与其他代数表示理论的联系等。

本项目为期四年，从2015 年1 月开始至2018 年12 月结束；依照既定研究计划，在李代数表示领域取得一系列研究成果，达到了预期研究效果。本项目首先对无扭仿射型Kac-Moody 代数构造了一大类不可约模，利用这些模，给出了无扭仿射李代数正根元素作用局部有限的不可约模的刻画及同构的条件。最终，我们给出了无扭仿射李代数最高权模和loop 模张量积的不可约性的充要条件，完全解决了这个30多年的公开问题。然后，我们把部分结论推广到了有扭的情况，即，给出了有扭仿射李代数正根元素作用局部有限的不可约模的刻画。同时，本项目还研究了许多和Kac-Moody代数相关的李代数的表示理论，如Virasoro 代数，Witt代数，Cartan 型李代数，Block型李代数，Heisenberg-Virasoro 代数，W代数W(2,2)，Virasoro-like代数，在这些李代数的U(h)-自由模，权空间无限维的不可约模，不可约非权模的构造，张量积模的不可约性等方面也取得一些成果。项目执行期间，项目组成员共发表或者被接收学术论文14篇，均为SCI检索，另有2 篇已经投稿。项目期间，一名项目组成员攻读博士学位，并获留学基金委资助赴加拿大进行联合培养；项目负责人指导硕士研究生10人，其中7人已顺利毕业，获得硕士学位。