Research in Ring Theory and Noncommutative Algebraic Geometry

环论与非交换代数几何研究

• 批准号: 0800948
• 负责人: Kenneth Goodearl
• 金额: $ 15.05万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800948&HistoricalAwards=false
• 关键词: Research; Ring; Theory; Noncommutative; Algebraic

The Principal Investigator proposes to continue his investigations into the structure of various classes of noncommutative rings, particularly quantum coordinate rings (quantized algebras offunctions) and related algebras, with a focus on geometric aspects of noncommutative rings and ring-theoretic aspects of noncommutative algebraic geometry. The proposed projects, located within noncommutative algebra, will build up the infrastructure of and interconnections among active parts of several related areas-- noncommutative algebraic geometry, quantum groups, Poisson algebras, and ring theory. Roughly speaking, the long-term goals framing many of these projects stem from noncommutative algebraic geometry, and the basic examples come from quantum groups, while many lines of approach are recruited from ring theory. A major new drive directing the PI's research on the subject is the investigation of Poisson analogs in semiclassical limits of phenomena known or expected to hold in quantum algebras. The parallels that have arisen so far provide strong motivation for the strategy of linking up the two sides of the coin; at this point, knowledge of one side allows the formulation of more precise conjectures for the other. A major goal is to tighten these connections.A pervasive theme in the mathematical study of geometric objects is that the properties of these objects are completely encoded in the functions on them, and are often more accessible via these functions than directly. Within algebraic geometry -- the study of geometric spaces defined by polynomial equations -- it is the polynomial functions on a space that determine it. These functions form a ring (a system endowed with compatible addition and multiplication operations) which is, moreover, commutative (fg = gf always). In the 1980s, researchers in the former Soviet Union, in the process of solving certain problems in theoretical quantum physics, discovered rings which appear to enjoy all the structure of rings of functions on geometric spaces, except that the multiplication is noncommutative. In honor of their origins in quantum theory, these rings are now called "quantized coordinate rings." It proved very useful to treat them as if they were rings of functions (except for the noncommutativity), and the guiding principle in their study became the search for "noncommutative versions of the geometry." Sufficiently many common phenomena (both geometric and algebraic) have been discovered in a wide range of quantized coordinate rings to lead one to conjecture that general, axiomatizable underpinnings within this class of rings are responsible for the parallels in their behavior. The main long-term thrust of the PI's research is to uncover such general structures and decode their geometric content. In the medium term, the proposal aims to extend the range of known shared phenomena within this class of rings, in order to gain better insight into their common base.

首席研究员建议继续他的调查到各种类型的非交换环的结构，特别是量子坐标环（量子代数offunctions）和相关的代数，重点是几何方面的非交换环和环理论方面的非交换代数几何。拟议的项目，位于非交换代数，将建立基础设施和几个相关领域的活跃部分之间的互连-非交换代数几何，量子群，泊松代数和环理论。粗略地说，许多这些项目的长期目标来自非交换代数几何，基本的例子来自量子群，而许多方法来自环理论。一个主要的新的驱动力指导PI的研究主题是调查泊松类似物的半经典极限的现象已知或预期举行的量子代数。迄今为止出现的相似之处为将硬币的两面联系起来的策略提供了强大的动力;在这一点上，对一面的了解可以为另一面制定更精确的解释。一个主要的目标是加强这些联系。在几何对象的数学研究中，一个普遍的主题是这些对象的属性完全编码在它们的函数中，并且通常比直接通过这些函数更容易访问。在代数几何中--研究由多项式方程定义的几何空间--是空间上的多项式函数决定了它。这些函数形成了一个环（一个具有相容的加法和乘法运算的系统），而且这个环是交换的（fg = gf总是）。在20世纪80年代，前苏联的研究人员在解决理论量子物理学中的某些问题的过程中，发现了似乎享有几何空间上函数环的所有结构的环，除了乘法是非交换的。为了荣誉它们起源于量子理论，这些环现在被称为“量子化坐标环”。事实证明，把它们当作函数环来对待是非常有用的（除了非交换性），他们研究的指导原则变成了寻找“非交换形式的几何”。在广泛的量子化坐标环中，已经发现了足够多的常见现象（几何和代数），这导致人们猜想，这类环中的一般性，可公理化的基础是它们行为中的相似之处。PI研究的主要长期目标是揭示这种一般结构并解码其几何内容。从中期来看，该提案旨在扩大这类环内已知共有现象的范围，以便更好地了解它们的共同基础。