Mathematical Sciences: Representation Theory of Lie Algebras

Enright将研究在交换环上定义的李代数的表示理论，该交换环上有一个显著的极大理想，其馀域为复数。特别令人感兴趣的是，在环上定义的李代数水平上的信息，是否可以通过从环到剩余域的通道，为复数上定义的更传统的李代数提供重要的信息。李群理论是为了纪念挪威数学家索夫斯·李而命名的，它一直是20世纪数学的主要主题之一。作为利用系统中固有对称性的数学工具，李群表示理论对数学本身，特别是分析和数论，以及理论物理，特别是量子力学和基本粒子物理产生了深远的影响。

Enright will study the representation theory of Lie algebras defined over a commutative ring in a setting where a distinguished maximal ideal is proscribed for which the residue field is the complex numbers. Of particular interest is whether information at the level of Lie algebras defined over rings yield important information for the more traditional Lie algebras defined over the complex numbers by passage from the ring to the residue field. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.