Research in Ring Theory and Noncommutative Algebraic Geometry

首席研究员建议继续研究各类非交换环的结构，特别是量子坐标环（量子化代数函数）和相关代数，重点研究非交换环的几何方面和非交换代数几何的环论方面。拟议的项目位于非交换代数中，将建立几个相关领域的基础设施和相互联系，这些领域包括非交换代数几何、量子群、泊松代数和环理论。粗略地说，构建这些项目的长期目标源于非交换代数几何，基本的例子来自量子群，而许多方法来自环理论。指导PI研究这一课题的一个主要新动力是在量子代数中已知或预期存在的现象的半经典极限中对泊松类似物的研究。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，对一方的了解允许对另一方进行更精确的推测。一个主要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，并且通常通过这些函数比直接访问更容易。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，这个环是可交换的（fg = gf总是）。20世纪80年代，前苏联的研究人员在解决理论量子物理中的某些问题的过程中，发现环似乎具有几何空间上函数环的所有结构，只是乘法是不可交换的。为了纪念它们在量子理论中的起源，这些环现在被称为“量子化坐标环”。事实证明，将它们视为函数环是非常有用的（除了非交换性），他们研究的指导原则变成了寻找“几何的非交换版本”。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），这使得人们推测，这类环中一般的、公理化的基础是它们行为相似的原因。PI研究的主要长期目标是揭示这些一般结构并解码它们的几何内容。从中期来看，该提案旨在扩大这类环内已知共享现象的范围，以便更好地了解它们的共同基础。

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.