Simple smooth representations of Lie algebras

撒谎 代数理论在许多学科中得到越来越广泛的应用 数学和物理，包括结合代数、代数几何、几何 表示法，顶点算子代数，偏微分方程式，代数 组合学、弦论、保形场理论、孤子理论。此外 李代数理论在许多学科中都很有用，具有内在的吸引力， 在其基础上结合了极大的深度和令人满意的完整程度 理论。这一理论及其理论仍存在许多有趣的问题。 申请。例如，一个重要的问题是如何将不同的 各种李代数的不可约表示类。 表示法 李代数的理论还远远没有发展得很好。一个原因是它 一般认为不可能将所有不可约模归类为非平凡模。 狄克斯米尔所指出的李代数。到目前为止，只有二维非阿贝尔 李代数，三维单李代数，三维海森堡 代数和它们的一些变形都有这样的分类。 这个 本研究的主要目的是研究一些重要的不可约表示 物理中的李代数，如仿射Kac-Moody代数，Virasoro 代数、扭曲的Heisenberg-Virasoro代数和其他具有 有用的结构和应用(例如，超椭圆李代数和Witt代数)。 另一个 我的计划的目标是将李代数理论应用于研究 代数几何中代数曲线的同构和自同构 多项式环的自同构的研究，Aut(C[x1，x2， ...，xn])，这是n&gt；2在交换中长期存在的问题 代数。 更多 准确地说，未来五年，我计划开展以下工作： 调查。 1. 仿射Kac-Moody代数的不可约光滑表示的分类 仿射Kac-Moody的特殊不可约限制表示 代数； 2. 有限维权的不可约权表示的分类 Witt代数W+n的空间，并确定更高的高度(&gt；1) 上的简单光滑模

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