Mathematical Sciences: Lie Algebras, Lie Superalgebras, and Quantum Groups

这个奖是关于李代数理论的研究。其中一名主要研究人员将研究素数特征的简单和半简单李超代数的分类。他还将研究顶点算子，特别是某些模块的显式构造，以及某些Hopf代数。第二名首席研究员将研究量子群的表示和量子Yang-Baxter方程的相关解。她计划在单位根的仿射Kac-Moody李代数的量子包络代数的情况下工作。她还计划描述最小循环表示，特别是证明最小维数等于相应李群最小轨道的维数。第三位首席研究员将研究李超代数的表示理论的各个方面。他将研究经典李超代数和扭曲d模上的泛模；经典李超代数的有限维非典型模cartan型经典李超代数的表示半单李超代数的表示以及Kac-Moody Lie超代数的表示。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。李群和李代数在球面运动分析中是很重要的。

This award concerns research on the theory of Lie algebras. One of the principal investigators will work on the classification of simple and semisimple Lie superalgebras of prime characteristic. He will also work on vertex operators, specifically on explicit construction of certain modules, as well as on certain Hopf algebras. A second principal investigator will study representations of quantum groups and associated solutions of the quantum Yang-Baxter equations. She plans work on the case of quantum enveloping algebras of affine Kac-Moody Lie algebras at roots of unity. She also plans to describe minimal cyclic representations, and in particular to show that the minimum dimension equals the dimension of the minimal orbit of the corresponding Lie group. A third principal investigator will study various aspects of the representation theory of Lie superalgebras. He will work on generic modules over classical Lie superalgebras and twisted D-modules; finite-dimensional atypical modules of classical Lie superalgebras; representations of Cartan-type classical Lie superalgebras; representations of semisimple Lie superalgebras; and representations of Kac-Moody Lie superalgebras. 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 spherical motion.