单位根处的量子超代数

作为量子群的重要推广，量子超代数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单位根处的量子超代数和量子仿射超代数的结构和表示理论，主要包括：研究如何利用量子仿射超代数的生成元的交换子构造出这些超代数的q-根向量；研究量子超代数和量子仿射超代数的整形式及其表现，尝试构造出量子超代数和量子仿射超代数的Lusztig整形式；研究量子超代数上的Yamane同构在q-根向量的divided power上的作用，并探索Yamane同构是否满足相应的辫子群关系；研究量子超代数在单位根处的小量子超代数的存在性并尝试建立量子Frobenius超映射和张量积定理；研究量子超代数特别是量子queer超代数在单位根处的有限维不可约表示的分类；研究量子超代数的Harish-Chandra同态及中心子代数的结构。

As an important generalization of quantum groups, quantum superalgebras have applications in many branches of mathematics and physics. Based on what we have done before in the field of representation theory of quantum superalgebras, we will study the structure and representation theory of quantum superalgebras and quantum affine superalgebras at roots of unity. We will study how to use the commutators of the Chevalley generators of quantum affine superalgebras to construct the q-root vectors in these superalgebras, as a generalization of the approach Yamane used to construct the q-root vectors in quantum superalgebras. As an application, we will study the analog of Lusztig's integral form for quantum superalgebras and quantum affine superalgebras. The Yamane isomorphims on quantum superalgebras will also be explored and in particular we will focus on its action on the divided powers of q-root vectors to check whether these isomorphisms satisfy the braid group relations. Then we shall also explore the existence of the small quantum superalgebras, quantum Frobenius map and tensor product theorem for quantum superalgebras. We will also study the classification of finite dimensional irreducible representations of quantum superalgebras such as quantum queer superalgebras at roots of unity. Finally we will explore the Harish-Chandra homomorphism for quantum superalgebras and the structure of their centers.