Lie Superalgebra Representations: Algebraic and Geometric Methods

近年来，李超代数的表示理论有了几个重要的进展。其中之一是Serganova将关于非典型有限维gl(m+ne)-模的特征的Kac问题简化为类似Vogan函子的半简单猜想。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。最后是由Serganova和主要研究者提出的任意有限维李超代数的三角分解理论和一般不可约最高权模的相关理论。该奖项支持的研究涉及与这些发展自然相关的五个不同主题：任意有限维李超代数上的有限维泛模的几何；有限维李超代数上的任意泛型模；经典李超代数上的特殊（非泛型和非典型）模；Kac-Moody超代数上的一般可积模量子超代数。在大多数情况下，计划使用代数和几何方法的组合。这项研究涉及一个叫做李代数的数学对象。李代数是由另一个对象李群产生的。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到精子运动分析的领域是重要的。

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.