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; 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

说谎 代数理论已在许多分支中越来越广泛地使用 数学和物理学，包括联想代数，代数几何形状，几何形状 表示形式，顶点操作员代数，部分微分方程，代数 组合学，弦理论，保形场理论，孤子理论。除了 谎言代数理论在许多主题中有用，本质上是吸引人的， 将深度和令人满意的完整性结合在一起 理论。该理论仍然有许多有趣的问题及其 申请。例如，一个重要的问题是如何对不同 各种谎言代数的不可还原表示。 表示 谎言代数的理论远非良好的发展。原因之一是 通常认为不可能对任何不平凡的所有不可还原模块进行分类 lie代数为dixmier指出。到目前为止，只有二维非亚伯利亚人 Lie代数，三维简单的谎言代数，三维的海森伯格 代数及其某些变形具有这样的分类。 这 我的研究的主要目的是研究一些不可约的表示 在物理学中介绍代数，例如Aggine Kac-Moody代数，Virasoro 代数，扭曲的海森堡 - 维拉索罗代数，其他代数 有用的结构和应用（例如，超椭圆形谎言代数和Witt代数）。 其他 我程序的目的是将谎言代数理论应用于研究 代数几何形状中代数曲线的同构和自动形态， 多项式环的自动形态的研究，AUT（C [x1，x2， ...，xn]），这是n> 2的长期问题 代数。 更多的 确切地说，在接下来的五年中，我计划进行以下 调查。 1。 分类为Aggine Kac-Moody代数的不可还原平滑表示， 特别的，不可证明的限制表示仿射kac-moody 代数； 2。 将不可还能的重量表示与有限的尺寸重量分类 WITT代数W+N的空间，并确定更高的高度（> 1） 简单平滑模块