作为量子群的重要推广，量子超群在数学和物理的很多分支有着广泛的应用。本项目力图在前期已有的工作基础上进一步研究量子线性超群和量子奇异超群以及相关的超代数如量子舒尔超代数和量子奇异舒尔超代数的表示理论，主要包括：研究量子线性超群和量子奇异超群的Fock 空间，尝试构造这些空间的典范基以及建立与Hecke代数和Hecke-Clifford超代数的Kazhdan-Lusztig多项式的联系；研究量子奇异超群的Beilinson-Lusztig-MacPherson实现理论和典范基理论，力图构造出量子奇异超群的典范基；建立量子舒尔超代数和量子奇异舒尔超代数的Z-分次结构；研究量子奇异舒尔超代数的胞腔结构；研究Hecke代数在经典李超代数的两两正交的根构成的集合所张成的空间上的作用及该空间的典范基。

As an important generalization of quantum groups, quantum supergroups have applications in many aspects in Mathematics and Physics. We will study the representation theory of quantum linear supergroups and quantum queer supergroups as well as related superalgebras including quantum Schur superalgebras and quantum queer Schur superalgebras. We will construct the canonical bases of the Fock of quantum linear supergoups and quantum queer supergroups and explore its connection with the Kazhdan-Lusztig polynomials of Hecke algebras and Hecke-Clifford superalgebras. We shall also study the Beilinson-Lusztig-MacPherson realization and canonical basis of quantum queer supergroups. We will provide the cellular structure and Z-grading on quantum queer Schur superalgebras. We will also explore the action of Hecke algebras on the space spanned by the set of orthogonal roots of classical Lie superalgebras as well as the canonical basis of this space.