量子群是数学中心课题之一的李理论中涵盖甚广的研究课题，其内容涉及到数学和物理的关键分支。本研究计划旨在对量子Virasoro代数、量子W代数、量子环面代数、Yangian代数、顶点算子代数和量子顶点代数、Quiver Hecke代数的表示和结构进行深入研究，以期洞悉他们之间的内在关系。具体的我们将通过确定q-Virasoro代数一类表示上的所有奇异向量来澄清它和量子仿射代数的关系，将构造BCD型Yangian代数并利用融合系方法研究其表示。在量子高维仿射代数方面，将确定所有零度为2的量子代数及其表示。我们将构造新的量子顶点代数并研究其表示。为了呼应量子群理论，我们将彻底证明顶点算子代数的有理性猜想，并研究相关模的张量分解。同时我们也将研究箭筒Hecke代数的中心特征标和分解矩阵，确定BMW代数的分次性。通过这些代数的表示可以用来研究许多以前看似无关的数学问题，更好理解量子群和对称性的关系

Quantum groups are an important class of algebras in Lie theory, a core mathematical interdisciplinery subject that has profound applications to mathematics and physics. This project intends to conduct far-reaching investigations on quantum Virasoro algebras, quantum W-algebras, quantum toroidal algebras, Yangian algebras, vertex operator algebras and quantum vertex algebras, quiver Hecke algebras, and their mutual relations as well as their representations. Specifically we will determine singular vectors of certain representation of the q-Virasoro algebra to clarify its relation with quantum affine algebras. We plan to construct the BCD type Yangians and use the fusion procedure to study their representations. In quantum extended algebras we will determine all such algebras of nullity 2 as well as their representations. New quantum vertex algebras will be constructed and their representations will be studied. We will solve the rationality conjecture in the VOA theory. Finally we will study the central characters of the quiver Hecke algebras and fix their decomposition matrices, and we also study the gradation property of the BMW algebra. Research has shown that these types of infinite-dimensional algebras can reveal unexpected relations among different problems in mathematics and help us understand how quantum groups serve as symmetry in the nature.