Linear algebraic groups and related topics in algebra

线性代数群和代数中的相关主题

• 批准号: 1201542
• 负责人: Parimala Raman
• 金额: $ 14.13万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201542&HistoricalAwards=false
• 关键词: Linear; algebraic; groups; related; topics

The PI proposes to exploit recently developed techniques in the study of projective homogeneous varieties to pursue several questions in the theory of semisimple algebraic groups over an arbitrary field, their Galois cohomology, and their cohomological invariants. The new techniques have already produced some extremely strong results and the investigator proposes to push them further. Previous work on these topics has had applications to other areas of mathematics and to physics, and further such applications are expected.The family of semisimple groups includes familiar matrix groups like special linear and special orthogonal groups. These groups appear in many areas of mathematics, and may be viewed as an essential outgrowth of the linear algebra developed in the early 1800s and now taught to undergraduates. The groups became prominent objects of mathematical interest in the late 1800s via Sophus Lie's famous general theory of Lie groups. In algebra, the notion of semisimple group unifies various historically distinct areas of study. For example, it connects Jordan algebras -- discovered by physicists -- with quadratic forms and division algebras.

PI建议利用最近开发的技术在研究射影齐性品种追求几个问题的理论半单代数群在任意领域，他们的伽罗瓦上同调，和他们的上同调不变量。 新技术已经产生了一些非常强有力的结果，研究人员建议进一步推动它们。 以前的工作对这些主题已经应用到其他领域的数学和物理，并进一步这样的应用程序是expected.家庭的半单群包括熟悉的矩阵群，如特殊的线性和特殊的正交群。这些群出现在数学的许多领域，并且可以被看作是19世纪早期发展起来的线性代数的重要产物，现在教授给本科生。在1800年代后期，通过Sophus Lie著名的李群一般理论，李群成为数学兴趣的突出对象。在代数中，半单群的概念统一了历史上不同的研究领域。例如，它将物理学家发现的Jordan代数与二次型和除法代数联系起来。