仿射李超代数和超Yangian代数的表示

仿射李（超）代数在高维情形的推广和量子化近年来受到数学家们的广泛关注。量子环面上的李超代数和超Yangian代数分别是扩张仿射李代数和仿射量子超群中重要的研究对象。它们为Yang-Baxter方程和自旋链模型等理论物理的研究提供有效的工具，并给有限维和仿射李（超）代数的研究带来了新的视角。本项目拟对量子环面上的李超代数和超Yangian代数的表示进行研究，致力于解决四个方面的问题：（1）量子环面上的一般线性李超代数和正交辛李超代数的Weyl模及其不可约性；（2）正交辛和Queer超Yangian代数的有限维不可约表示的分类；（3）超Yangian代数基本表示的张量表示的不可约性；（4）超Yangian Double顶点表示和自由场表示的构造和不可约性。对这些问题的研究将进一步丰富这些超代数的表示理论，并为更深入研究仿射李（超）代数及其量子化的表示理论奠定基础。

Higher rank generalizations and quantizations of affine Lie (super) algebras have attracted many mathematician's attention in recent years. Lie superalgebras over quantum tori and super-Yangians are important objects in extended affine Lie algebras and quantum supergroups respectively. They have provided useful tools for exploring Yang-Baxter equations and spin chain models in theoretical physics, and have also introduced new ideas and methods into the investigations on finite-dimensional or affine Lie superalgebras. Our project plans to study representations of Lie superalgebras over quantum tori and super-Yangians. We will primarily focus on the following four topics: (1) Weyl modules for general linear Lie superalgebras and ortho-symplectic Lie superalgebra over a quantum torus and their irreducibility; (2) Classifications of finite-dimensional irreducible representations for the queer and ortho-symplectic super-Yangians; (3) Irreducibility of the tensor product of fundamental representations for super-Yangians; (4) Vertex representations and free field representations of super-Yangian doubles. Our research will enrich the representation theory of these superalgebras and will inspire further investigations on representations of affine Lie (super)algebras and their quantizations.