Interactions between noncommutative algebra, algebraic geometry and representation theory

非交换代数、代数几何和表示论之间的相互作用

• 批准号: 1004306
• 负责人: Rajesh Kulkarni
• 金额: $ 15万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1004306&HistoricalAwards=false
• 关键词: Interactions; between; noncommutative; algebra; algebraic

The proposed research is at the intersection of noncommutative algebra, algebraic geometry and representation theory. The projects in the first direction concern noncommutative analogs of algebraic surfaces called maximal orders on surfaces. These are coherent sheaves of noncommutative algebras on surfaces whose generic stalk is a central simple algebra. The proposed projects related to maximal orders include studying coarse moduli spaces of terminal orders, orders on 3-folds, module and derived categories of certain terminal orders and Brauer groups of some K3 surfaces in characteristic p. The second direction of proposed projects concerns Clifford algebras of higher degree forms and their representations. The projects in this direction include using arithmetically Cohen-Macaulay bundles to construct moduli spaces of irreducible representations of Clifford algebras of ternary cubic forms, and using Clifford algebras of binary forms to construct maximal orders on surfaces. The third direction of proposed projects concerns computations of Lefschetz numbers for Hecke correspondences of certain Shimura varieties. Over the past few decades there have been increasing and fruitful interactions between various areas of mathematics and theoretical physics. One important thread in these connections has been algebraic geometry, a very old subject that dates back at least to ancient Greece. Algebraic geometry is a subject in which solutions of many variable polynomial equations are studied as geometric objects. The proposed projects use tools from algebraic geometry to solve problems in noncommutative algebra. In noncommutative algebra, the main objects of study are polynomials in which the xy is not the same as yx. Such noncommutative algebras are of interest to physicists as well. A great deal of progress has been achieved in the past 15 years in understanding such algebras and the projects in this proposal will advance this knowledge. The proposed projects in another direction lie at the intersection of number theory (where number systems are studied), representation theory (where symmetries are studied), and topology. In topology, properties of spaces are studied which do not change under elastic deformations such as twisting and stretching. The spaces to be studied in this project encode number theoretic information and possess various kinds of symmetries.

提出的研究是非对易代数、代数几何和表示论的交集。第一个方向的投影涉及被称为曲面上的最大阶的代数曲面的非交换类似。这是曲面上的非对易代数的凝聚层，其通用柄是中心单代数。与极大阶有关的项目包括研究特征p中某些K3曲面的端阶、3-折叠上的阶、模和导子范畴的粗模空间以及某些K3曲面的Brauer群。第二个方向是高次形式的Clifford代数及其表示。在这个方向上的计划包括利用算术Cohen-Macaulay丛来构造三值三次形Clifford代数的不可约表示的模空间，以及使用二元形式的Clifford代数来构造曲面上的最大阶。建议项目的第三个方向涉及某些下村变种的Hecke对应的Lefschetz数的计算。在过去的几十年里，数学和理论物理的各个领域之间的互动越来越多，成果也越来越多。这些联系中的一条重要线索是代数几何，这是一个非常古老的学科，至少可以追溯到古希腊。代数几何是一门把许多变量多项式方程的解作为几何对象来研究的学科。建议的项目使用代数几何中的工具来解决非对易代数中的问题。在非交换代数中，主要的研究对象是XY与YX不同的多项式。这样的非交换代数也引起了物理学家的兴趣。在过去的15年里，在理解这种代数方面已经取得了很大的进展，本提案中的项目将促进这一知识的发展。在另一个方向上提出的项目位于数论(研究数字系统)、表示理论(研究对称性)和拓扑学的交叉点。在拓扑学中，研究了在扭转和拉伸等弹性变形下不变的空间的性质。本课题要研究的空间编码了数论信息，并具有各种对称性。