Regular Algebras

正则代数

• 批准号: 0900239
• 负责人: Michaela Vancliff
• 金额: $ 11.69万
• 依托单位: University of Texas at Arlington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900239&HistoricalAwards=false
• 关键词: Regular; Algebras

This award supports the research of Michaela Vancliff to work in non-commutative algebra, with special emphasis on problems arising from the theory of regular algebras and non-commutative algebraic geometry. She is interested in the graded-module category 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 of higher-dimensional linear schemes generalize the structure and role of point schemes. Under prior NSF support, in work with T. Cassidy, Vancliff produced algebro-geometric techniques that allow the easy construction of certain regular algebras (that generalize graded Clifford algebras) of any finite global dimension; naming such algebras graded skew Clifford algebras. Vancliff intends to study such regular algebras of global dimension four that have finitely many points and a one-parameter family of line modules as a step towards classifying the line schemes that arise for ``generic'' quadratic regular algebras of global dimension four. Her initial research with B. Shelton (under prior NSF support) suggests that such an algebra should have a line scheme that consists of exactly six elliptic curves, so if this is found to hold in general, then it would mimic the point scheme of generic quadratic regular algebras of global dimension three (where it is one elliptic curve).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. Often, the solutions cannot be found by experimentation, and often they are not numbers but are functions (e.g., differential operators or matrices), and so, in general, they do not commute. The science of seeking methods that find all solutions to any system of polynomial-style equations in non-commuting variables is called non-commutative algebra. To find the solutions, the main idea is as follows. One associates to such a system of equations an entity, called an ``algebra'', that encodes all the properties of the original equations. Associated to this algebra are ``modules'', and these encode all the properties of the solutions to the equations. So, in order to find all the solutions, one should find all the modules for the associated algebra. In many of the applications, the algebras that arise in this way tend to share certain properties satisfied by the polynomial ring; such algebras are called regular algebras and are the main focus of Vancliff's projects. One of the goals of non-commutative algebraic geometry, the subfield in which Vancliff works, is to use geometric techniques to find certain 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 of modules.

该奖项支持Michaela Vancliff在非交换代数方面的研究，特别强调正则代数和非交换代数几何理论所产生的问题。她感兴趣的分级模块类别视为一个几何空间，与某些分级模块发挥作用的几何对象。线性几何模块（点模块、线模块等）由所谓的线性方案参数化。Vancliff计划研究高维线性方案的结构和作用如何推广点方案的结构和作用。在NSF之前的支持下，与T. Cassidy，Vancliff生产代数几何技术，使容易建设某些经常代数（推广分次Clifford代数）的任何有限的全球层面;命名这样的代数分次斜Clifford代数。Vancliff打算研究这样的正规代数的全球维度4有许多点和一个参数家庭的线模块作为一个步骤分类线计划出现的“一般”二次正规代数的全球维度4。她和B最初的研究。谢尔顿（在先前的NSF支持下）建议这样的代数应该有一个由正好六条椭圆曲线组成的线方案，所以如果发现这在一般情况下成立，那么它将模仿全局维数为3的一般二次正则代数的点方案。（其中它是一条椭圆曲线）。多项式型方程组及其解在几乎每个科学领域都起着关键作用，例如统计力学、基本粒子物理学、量子力学、机器人学、晶体学、网络等。通常，不能通过实验找到解决方案，并且通常它们不是数字而是函数（例如，微分算子或矩阵），因此，一般来说，它们不交换。寻找非交换变量的任何多项式方程组的所有解的方法的科学称为非交换代数。为了找到解决方案，主要思想如下。人们把这样一个方程组与一个实体联系起来，称为“代数”，它把原始方程的所有性质都编码了。与这个代数相关联的是“模”，它们编码了方程解的所有性质。 所以，为了找到所有的解，我们应该找到相关代数的所有模。 在许多应用中，以这种方式产生的代数往往具有多项式环所满足的某些性质;这样的代数被称为正则代数，是Vancliff项目的主要焦点。其中一个目标的非交换代数几何，该子领域中的范克利夫工程，是使用几何技术，以找到某些模块（点模块，线模块等）的经常代数，然后使用这些模块找到模块给予解决方案的原始系统的方程。Vancliff的根本目标是改进这些几何技术，并更好地理解它们如何与模块类别的结构相关。