Simple smooth representations of Lie algebras

李代数的简单平滑表示

• 批准号: RGPIN-2020-04774
• 负责人: Zhao, Kaiming
• 金额: $ 1.53万
• 依托单位: Wilfrid Laurier University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717348
• 关键词: Simple; smooth; representations; Lie; algebras

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

谎言 代数理论在数学的许多分支中得到了越来越广泛的应用。 数学和物理，包括结合代数，代数几何，几何 表示，顶点算子代数，偏微分方程，代数 组合学，弦论，共形场论，孤立子理论。除了 李代数理论在许多学科中都很有用，因此具有内在的吸引力， 结合了很大的深度和令人满意的程度，在其基本的完整性， 理论该理论及其应用还存在许多有趣的问题， 应用.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 李代数的理论还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的不可约模进行分类。 李代数，由Dixlane指出。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 代数和它们的某些变形有这样的分类。 的 我研究的主要目的是研究一些重要的不可约表示 物理学中的李代数，如仿射Kac-Moody代数，Virasoro代数 代数，扭曲海森堡-Virasoro代数，以及其他李代数， 有用的结构和应用（例如，超椭圆李代数和Witt代数）。 另一 我的计划的目标是应用李代数理论的研究， 代数几何中代数曲线的同构和自同构， 多项式环Aut（C[x1，x2， ..., xn]），这是交换中n>2的长期存在的问题。 代数 更 准确地说，在未来五年，我计划开展以下工作 调查事务所 1. 分类仿射Kac-Moody代数的不可约光滑表示， 仿射Kac-Moody的特殊不可约限制表示 代数; 2. 有限维权不可约权表示的分类 Witt代数W+n的空间，并确定了高（>1） 单光滑模