Lie Superalgebra Representations: Algebraic and Geometric Methods

近年来，李超代数的表示理论有了一些本质的发展。其中之一是Serganova将关于非典型有限维gl(m+ne)-模的特征的Kac问题归结为类似于Vogan函子的函子的半单性猜想。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，给出了任意有限维李超代数的三角分解理论，以及Serganova和主要研究者发展的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通用模的几何；有限维李超代数上的任意通用模；经典李超代数上的特殊(非通用和非典型)模；Kac-Moody超代数上的通用可积模；以及量子超代数。在大多数情况下，计划使用代数和几何方法的组合。这项研究涉及一种名为李代数的数学对象。李代数起源于另一个叫做李群的物体。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球面运动分析的领域中很重要。

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.