Research in Ring Theory and Noncommutative Algebraic Geometry

提议题目：环理论和非交换代数几何研究首席研究员：k.r.g oodearl首席研究员建议继续研究各种非交换环的结构，特别是量子坐标环（量化代数函数）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理论方面。拟议的项目位于非交换代数中，将建立几个相关领域的活跃部分之间的基础设施和相互联系-环理论，非交换代数几何和量子群。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。重点将放在量化坐标环的素谱和原谱的代数和几何特征上。目标包括素理想的分类，素理想在自同构环下不变的有限条件，支持性质的验证，如auslander -正则性和cohen - macaulayness，以及量子化素数或原谱作为经典谱或谱变体的拓扑商的表示。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（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，发现了环，它似乎具有几何空间中函数环的所有结构，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，把它们当作函数环（除了非交换性）是非常有用的，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共同现象（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上的相似之处的原因。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中已知共享现象的范围，以便更好地了解它们的共同基础。

Proposal Title: Research in Ring Theory and Noncommutative AlgebraicGeometryPrincipal Investigator: K. R. GoodearlThe Principal Investigator proposes to continue his investigationsinto the structure of various classes of noncommutative rings,particularly quantum coordinate rings (quantized algebras offunctions) and related algebras, with a focus on geometric aspects ofnoncommutative rings and ring-theoretic aspects of noncommutativealgebraic geometry. The proposed projects, located withinnoncommutative algebra, will build up the infrastructure of andinterconnections among active parts of several related areas -- ringtheory, noncommutative algebraic geometry, and quantum groups. Roughlyspeaking, the long-term goals framing many of these projects stem fromnoncommutative algebraic geometry, and the basic examples come fromquantum groups, while many lines of approach are recruited from ringtheory. The main focus will be on algebraic and geometric features ofthe prime and primitive spectra of quantized coordinate rings. Thegoals include classification of prime ideals, finiteness conditionsfor prime ideals invariant under tori of automorphisms, verification ofsupporting properties such as Auslander-regularity andCohen-Macaulayness, and presentations of quantized prime or primitivespectra as topological quotients of classical spectra or varieties.A pervasive theme in the mathematical study of geometric objects isthat the properties of these objects are completely encoded in thefunctions on them, and are often more accessible via these functionsthan directly. Within algebraic geometry -- the study of geometricspaces defined by polynomial equations -- it is the polynomialfunctions on a space that determine it. These functions form a ring (asystem 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 solvingcertain problems in theoretical quantum physics, discovered ringswhich appear to enjoy all the structure of rings of functions ongeometric spaces, except that the multiplication is noncommutative. Inhonor of their origins in quantum theory, these rings are now called"quantized coordinate rings." It proved very useful to treat them asif they were rings of functions (except for the noncommutativity), andthe guiding principle in their study became the search for"noncommutative versions of the geometry." Sufficiently many commonphenomena (both geometric and algebraic) have been discovered in a widerange of quantized coordinate rings to lead one to conjecture thatgeneral, axiomatizable underpinnings within this class of rings areresponsible for the parallels in their behavior. The main long-termthrust of the PI's research is to uncover such general structures anddecode their geometric content. In the medium term, the proposal aimsto extend the range of known shared phenomena within this class ofrings, in order to gain better insight into their common base.