与仿射量子一般线性超群、量子queer超群、扭的仿射量子queer超群相关的Schur-Weyl理论和量子一般线性超群上的量子丛（超）代数

本项目将进一步丰富和完善Schur-Weyl理论，并将Beilinson、Lusztig和Macpherson关于量子一般线性群的BLM实现推广到仿射量子一般线性超群、量子queer超群和扭的仿射量子queer超群，具体包括：通过研究仿射A型Hecke代数、仿射量子Schur超代数和仿射量子一般线性超群的结构和表示理论，得到仿射量子一般线性超群BLM实现；通过研究Hecke-Clifford代数、量子queer超群、量子queer Schur超代数的结构和表示理论，得到量queer超群的实现；根据仿射Hecke-Clifford代数的自然表示，给出仿射量子queer Schur超代数的定义，并研究它的结构和表示理论以及与扭的仿射量子queer超群的关系，进而讨论扭的仿射量子queer超群的实现。. 本项目还将研究量子一般线性超群上的量子丛代数或量子丛超代数结构。

In this project, we will enrich and improve Schur-Weyl theory, and generalize the approach of Binlinson,Lusztig and MacPherson for realizing the quantum general linear group to affine quantum general linear supergroup, quantum queer supergroup and twisted affine quantum supergroup, which includes the aspects as follows: Through studying the structures and representation theory associated with affine Hecke algebra of type A, affine quantum Schur superalgebra and affine quantum general linear supergroup, we plan to obtain the realization of affine quantum general linear supergroup in BLM approach; Through studying the structure and representation theory associated with Hecke-Clifford algebra,quantum queer supergroup and quantum queer Schur superalgebra, we plan to achieve a realization of quantum queer supergroup; according to the natural representation of affine Hecke-Clifford algebra, we will define the affine quantum queer Schur superalgebras and study their structure and representation theory. We will also study their relations to twisted affine quantum queer supergroup, then we consider about the realization of twisted affine quantum queer supergroup.. In this project, we will study the quantum cluster algebra structure and quantum cluster superalgebra structure over quantum general linear supergroup.