Lie Superalgebra Representations: Algebraic and Geometric Methods

李超代数表示：代数和几何方法

• 批准号: 9500755
• 负责人: Ivan Penkov
• 金额: $ 4.65万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500755&HistoricalAwards=false
• 关键词: Lie; Superalgebra; Representations; Algebraic; Geometric

Recently there have been several essential developments in the representation theory of Lie superalgebras. One of them is Serganova's reduction of the Kac problem on characters of atypical finite-dimensional gl(m+ne)-modules to the semi-simplicity conjecture for a functor analogous to the Vogan functor. Another one is Kac and Wakimoto's discovery of a striking relationship between number theory and integrable representations of affine Lie superalgebras. Finally, there is the theory of triangular decomposition of arbitrary finite-dimensional Lie superalgebras and the related theory of generic irreducible highest weight modules developed by Serganova and the principal investigator. The research supported by this award is on five different topics related naturally to those developments: geometry of finite-dimensional generic modules over arbitrary finite-dimensional Lie superalgebras; arbitrary generic modules over finite-dimensional Lie superalgebras; special (both non-generic and atypical) modules over classical Lie superalgebras; generic integrable modules over Kac-Moody superalgebras; and quantum superalgebras. In most of the cases, it is planned to use a combination of algebraic and geometric methods. This research is concerned with a mathematical object called a Lie algebra. Lie algebras arise from another object called a Lie group. An example of a Lie group is the rotations of a sphere where one rotation is followed by another. Lie groups and Lie algebras are important in areas involving analysis of sperical motion.

最近，李超代数的表示理论有了一些重要的发展。 其中之一是Serganova的减少的卡茨问题的字符的非典型有限维gl（m+ne）-模的半单纯性猜想的函子类似的Vogan函子。 另一个是Kac和Wakimoto发现了数论和仿射李超代数的可积表示之间的惊人关系。 最后是任意有限维李超代数的三角分解理论，以及Serganova和主要研究者开发的一般不可约最高权模的相关理论。 该奖项所支持的研究是在五个不同的主题自然相关的发展：几何有限维通用模块在任意有限维李超代数;任意通用模块在有限维李超代数;特殊（非通用和非典型）模块在经典李超代数;通用可积模块在卡茨穆迪超代数;和量子超代数。 在大多数情况下，计划结合使用代数和几何方法。 这项研究涉及一个称为李代数的数学对象。 李代数起源于另一个称为李群的对象。 李群的一个例子是球体的旋转，其中一个旋转之后是另一个旋转。 李群和李代数在涉及分析球面运动的领域中是重要的。