Simple smooth representations of Lie algebras

李代数理论在数学和物理的许多分支中得到越来越广泛的应用，包括联想代数、代数几何、几何表示、顶点算子代数、偏微分方程、代数组合、弦论、共形场论、孤子理论等。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。该理论及其应用仍存在许多有趣的问题。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。李代数的表示理论还远远没有得到很好的发展。一个原因是一般认为不可能对任何非平凡李代数上的所有不可约模进行分类，如Dixmier所指出的那样。到目前为止，只有二维非阿贝尔李代数、三维简单李代数、三维海森堡代数和它们的一些变形有这样的分类。主要研究物理中重要李代数的一些不可约表示，如仿射Kac-Moody代数、Virasoro代数、扭曲Heisenberg-Virasoro代数，以及其他具有有用结构和应用的李代数（如超椭圆李代数和Witt代数）。我的项目的另一个目标是将李代数理论应用于代数几何中代数曲线的同构和自同构的研究，以及多项式环的自同构的研究，Aut(C[x1, x2，…， xn])，这是交换代数中一个长期存在的问题。更确切地说，在未来的五年里，我计划进行以下调查。1. 分类仿射Kac-Moody代数的不可约光滑表示，特别是仿射Kac-Moody代数的不可约限制表示2. 对Witt代数W+n有限维权空间的不可约权表示进行分类，并确定W+n上较高高度（>1）的简单光滑模；3. 构造新的简单加权或非加权sln+1模块，并对已知简单sln+1模块的某些类进行了刻画；4. 利用Heisenberg代数的简单表示在Cn和G2型有限维李代数上构造新的简单权模或非权模；5. 发展krichhever - novikov高属代数理论，研究代数几何中代数曲线的同构和自同构。这些问题的解决将有利于物理和数学的研究。

Lie algebra theory has become more and more widely used in many branches of mathematics and physics, including associative algebra, algebraic geometry, geometric representation, vertex operator algebra, partial differential equations, 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. There are still many interesting problems for the theory and its applications. For example, one important problem is how to classify different classes of irreducible representations for various Lie algebras. Representation theory for Lie algebras is far from being well developed. One reason is that it is generally considered impossible to classify all irreducible modules over any nontrivial Lie algebras as pointed out by Dixmier. So far, only the 2-dimensional non-abelian Lie algebra, the 3-dimensional simple Lie algebra, the 3-dimensional Heisenberg algebra and some of their deformations have such classifications. The main objective of my research is to study some irreducible representations for important Lie algebras in physics, such as affine Kac-Moody algebras, the Virasoro algebra, the twisted Heisenberg-Virasoro algebra, and other Lie algebras with useful structure and applications (for example, super-elliptic Lie algebras and Witt algebras).   Another objective of my program is to apply Lie algebra theory to the study of isomorphisms and automorphisms of algebraic curves in algebraic geometry, and to the study of automorphisms of polynomial rings, Aut(C[x1, x2, ..., xn]), which is a long standing problem for n>2 in commutative algebra. More precisely, in the next five years, I plan to carry out the following investigations. 1. Classify irreducible smooth representations for affine Kac-Moody algebras, in particular, irreducible restricted representations for affine Kac-Moody algebras; 2. Classify irreducible weight representations with finite dimensional weight spaces for Witt algebras W+n , and determine higher height (>1) simple smooth modules over W+n ; 3. Construct new simple weight or non-weight sln+1-modules and characterize some classes of known simple sln+1-modules; 4. Use simple representations of Heisenberg algebras to construct new simple weight or non-weight modules over the finite dimensional Lie algebras of type Cn and G2; 5. Develop Krichever-Novikov algebra theory of higher genus to study isomorphisms and automorphisms of algebraic curves in algebraic geometry. Solutions to these problems will benefit studies in both physics and mathematics.