Weight representations of lie algebras

李代数理论在数学、物理和金融的许多分支中得到越来越广泛的应用。联想代数理论、拓扑学、微分几何、代数组合学、二维统计模型理论、弦理论、共形场理论、孤子理论和其他数学物理都使用了李代数理论。除了在许多学科中都很有用外，李代数理论本身就很有吸引力，它的基础理论具有很强的深度和令人满意的完备性。它也提出了许多有趣和重要的问题。例如，如何对不同李代数的各种表示进行分类，并找出它们的特征公式。本研究的主要目的是研究物理学中一些重要李代数（如Kac-Moddy代数、广义Virasoro代数和扭曲Heisenberg-Virasoro代数）和其他一些具有有用结构的李代数（如量子环面李代数和广义Cartan型李代数）的良好表示。这些问题的解决将推动物理学和数学的理论研究。

Lie algebra theory has become more and more widely used in many branches of mathematics, physics, and finance. Associative algebra theory, topology, differential geometry, algebraic combinatorics, theory of 2-dimensional statistical models, string theory, conformal field theory, soliton theory, and other mathematical physics all use Lie algebra theory. Besides being useful in many subjects, Lie algebra theory is inherently attractive, combining a great depth and a satisfying degree of completeness in its basic theory. It also presents many interesting and important problems. For example, how to classify various representations for different Lie algebras and find out their character formulas. The main objective of my research is to study good representations for some important Lie algebras from physics (such as Kac-Moddy algebras, generalized Virasoro algebras and the twisted Heisenberg-Virasoro algebra), and for some other Lie algebras with useful structure (for example, quantum torus Lie algebras and generalized Cartan type Lie algebras). Solutions to these problems will advance theoretical studies in both physics and mathematics.