Irreducible weight and non-weight representations of some Lie algebras

李代数理论在数学和物理的许多分支中得到了越来越广泛的应用，包括结合代数、代数几何、拓扑学、微分几何、微分方程、代数组合学、弦论、保形场理论、孤子理论等。除了在许多学科中有用之外，李代数理论本身就很有吸引力，它的基本理论结合了巨大的深度和令人满意的完备性。同时也为该理论及其应用提出了许多有趣的问题。例如，一个重要的问题是如何对各种李代数的不可约表示进行分类。本研究的主要目的是研究物理上重要的李代数(如Kac-Mody代数、Virasoro代数和扭曲的Heisenberg-Virasoro代数)以及其他具有有用结构的李代数(如超椭圆李代数、量子环面李代数和Witt代数)的一些不可约表示。这些问题的解决将有利于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 and physics, including, associative algebra, algebraic geometry, topology, differential geometry, differential equation, algebraic combinatorics, string theory, conformal field theory, soliton 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 problems for the theory and its applications. For example, one important problem is how to classify irreducible representations for various Lie algebras. The main objective of my research is to study some irreducible representations for important Lie algebras from physics (such as Kac-Moddy algebras, the Virasoro algebra, and the twisted Heisenberg-Virasoro algebra), and for other Lie algebras with useful structure (for example, super-elliptic Lie algebras, quantum torus Lie algebras and Witt algebras). Solutions to these problems will benefit theoretical studies in both physics and mathematics.