Regular Algebras

正则代数

• 批准号: 0200757
• 负责人: Michaela Vancliff
• 金额: $ 9.86万
• 依托单位: University of Texas at Arlington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-15 至 2006-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200757&HistoricalAwards=false
• 关键词: Regular; Algebras

This award supports the research of Michaela Vancliff to work innon-commutative algebra, with special emphasis on problems arising from the theory of regular algebras and non-commutative algebraic geometry. Vancliff's main objective is to add to the body of knowledgeon regular algebras and to further the existing geometric techniques.In particular, she is interested in the graded-module category of suchan algebra viewed as a geometric space, with certain graded modules playing the role of geometric objects. The linear geometric modules (point modules, line modules, etc) are parametrized by so-called linear schemes. Vancliff plans to study how the structure and role ofhigher-dimensional linear schemes generalize the structure and role ofpoint schemes. She is also interested in connections between this typeof geometric analysis and that of various Poisson-geometric structures. Some of these activities entail the development of fundamental computational algorithms, and their implementation using a computer-algebra package, in order to enable explicit computation oflinear schemes.Systems of polynomial-style equations and their solutions play a critical role in almost every scientific field, such as statistical mechanics, elementary-particle physics, quantum mechanics, robotics, crystallography, networking, etc. The solutions are often results thatcannot be found by experimentation nor other methods, and often they are not numbers but are functions (e.g., differential operators or matrices), and so, in general, they do not commute. The business of seeking methods that find all solutions to any system of polynomial-style equations in non-commuting variables is called non-commutative algebra. The main idea to find the solutions is as follows. One associates to such a system of equations an abstract object called an algebra, which encodes all the properties of the system. To the algebra is associated abstract objects called modules,and these encode all the properties of the solutions. Hence, in orderto find all the solutions, one should find and understand all the modules for the associated algebra. In many of the applications, thealgebras that arise in this way tend to share certain properties; theyare called regular algebras and are the main focus of Vancliff'sprojects. One of the goals of non-commutative algebraic geometry, thesubfield in which Vancliff works, is to use geometric techniques to find some of the modules (point modules, line modules, etc) of the regular algebra, and then to use those modules to find the modules giving the solutions to the original system of equations. Vancliff's underlying goal is to improve on these geometric techniques and to understand better how they relate to the structure of the category ofmodules.

该奖项支持Michaela Vancliff致力于非交换代数的研究，特别强调由正则代数理论和非交换代数几何产生的问题。Vancliff的主要目标是增加正则代数的知识体系，并进一步发展现有的几何技巧，特别是她对被视为几何空间的这样的代数的分次模范畴感兴趣，其中某些分次模扮演几何对象的角色。通过所谓的线性方案将线性几何模块(点模块、线模块等)参数化。Vancliff计划研究高维线性格式的结构和作用如何泛化点格式的结构和作用。她还对这种类型的几何分析和各种泊松几何结构之间的联系感兴趣。其中一些活动需要开发基本的计算算法，并使用计算机代数包来实现它们，以便能够显式地计算线性模式。多项式形式的方程组及其解在几乎所有科学领域中都扮演着关键的角色，例如统计力学、基本粒子物理、量子力学、机器人、结晶学、网络等。解通常是通过实验或其他方法找不到的结果，并且通常不是数字而是函数(例如，微分算子或矩阵)，因此，一般而言，它们不是可交换的。在非对易变量中寻找任何多项式型方程组的所有解的方法被称为非对易代数。寻找解决方案的主要思路如下。人们将这样一个方程系统与一个称为代数的抽象对象联系在一起，它编码了该系统的所有性质。与代数相关的是称为模块的抽象对象，这些抽象对象编码了解的所有属性。因此，为了找到所有的解，一个人应该找到并理解相关代数的所有模块。在许多应用中，以这种方式产生的代数倾向于共享某些性质；它们被称为正则代数，并且是Vancliff项目的主要焦点。非对易代数几何是Vancliff工作的一个子域，它的目标之一是利用几何技巧找到正则代数的一些模(点模、线模等)，然后用这些模找到给出原始方程组的解的模。Vancliff的基本目标是改进这些几何技术，并更好地了解它们与模块范畴的结构之间的关系。