箭图表示理论以及Hall代数为李理论与量子群提供了新的视角。Fock空间表示在李代数、量子群和Hecke代数的表示理论等领域中有重要的应用。本项目拟研究仿射箭图的Hall代数的表示与典范基，并由此为量子仿射代数的Fock空间表示建立新的理论框架。同时构造量子线性超群的Fock空间的典范基。我们拟开展以下三个方面的研究；(1)构造仿射箭图Hall代数的generic子代数及它们的典范基，研究其Drinfeld double的最高权表示，并利用KLR-代数给出相关结构的范畴化；(2)将量子仿射代数的Fock空间表示扩展为上述generic子代数的表示，研究其不可约性、典范基、范畴化及其在表示论中的进一步应用；(3)构造量子线性超群的Fock空间的典范基，并研究这组基与A型Hecke代数的Kazhdan-Lusztig多项式的关系。

Representation theory of quivers and Hall algebras provide a new perspective for Lie theory and quantum groups. The Fock space representations have important applications in the representation theory of Lie algebras, quantum groups and Hecke algebras, etc. This project plans to study representations of Hall algebras of affine quivers and their canonical bases in order to set up a new framework for Fock space representations of quantum affine algebras，and construct the canonical basis of Fock spaces for quantum linear supergroups as well. We will focus on the following three topics: (1) Construct generic subalgebras of Hall algebras of affine quivers and their canonical bases, study highest weight representations of their Drinfeld doubles, and categorify these structures in terms of KLR algebras; (2) Extend the Fock space representations of quantum affine algebras to representations of the algebras constructed above, study their irreducibility, canonical basis, categorification and their further applications in representation theory; (3) Construct the canonical basis of Fock spaces for quantum linear supergroups, and establish a relationship between this basis and Kazhdan-Lusztig basis for Hecke algebras of type A.