Research in Ring Theory and Noncommutative Algebraic Geometry

环论与非交换代数几何研究

• 批准号: 1601184
• 负责人: Kenneth Goodearl
• 金额: $ 22.37万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601184&HistoricalAwards=false
• 关键词: Research; Ring; Theory; Noncommutative; Algebraic

The central aim of this research project is the study of various classes of quantum algebras (particularly quantized algebras of functions), with strong focus on both algebraic structure and geometric aspects. These algebras were discovered in the 1980s in the process of solving problems in theoretical quantum physics. They are mathematical systems that model rings of functions on geometric structures as in algebraic geometry (the study of geometric spaces defined by polynomial equations) except that the multiplication is noncommutative. The internal structure of quantum algebras will be intensively investigated, particularly those substructures that correspond to points and geometric subspaces in the classical analogs, together with relations between quantum and classical algebras that are connected via a semiclassical limit (a process in which the noncommutativity tends to zero). These commutative semiclassical limit algebras carry a remnant of the noncommutativity of the associated quantum algebras, recorded in a bilinear form called a Poisson bracket. The work will also pursue the closely related questions about Poisson algebras, as well as the classical-quantum interplay between Poisson semiclassical limits and their quantum counterparts. The research aims both to advance, and to tighten the links among, several highly active mathematical areas -- quantum groups, Poisson algebra, noncommutative algebraic geometry, and cluster theory. In particular, the work will furnish tools to pass insights from one area to the others. Specific projects include the exploitation of cluster algebra structures on large classes of quantum and Poisson algebras towards the ideal theory of these algebras. A related effort will go into the construction of more viable bridges over which to shift information back and forth between quantum algebras and their semiclassical limits. Along this line, the investigator conjectures that the prime and primitive spectra of a quantum algebra are homeomorphic to the Poisson-prime and Poisson-primitive spectra of a semiclassical limit, and he will devote major effort towards a confirmation. Other projects target the classical-quantum interplay. The goal is to develop noncommutative versions of properties enjoyed by commutative coordinate rings, prominent among them being catenarity of their prime spectra. The work will explore the conjecture that all quantized coordinate rings enjoy catenarity. These efforts are related to the goal of elucidating the global topological structures of the prime and primitive spectra of quantum algebras. This is aimed at gluing the (already secured) classical pieces of these spectra via auxiliary spaces and maps that are not required in the classical setting. Establishing a specific, detailed format for this gluing and auxiliary data, together with similar, matching formats for the spectra of quantum algebras and semiclassical limits, would yield explicit homeomorphisms between these spectra, and thus establish the conjectured quantum-Poisson interplay.

该研究项目的中心目标是研究各种量子代数（特别是函数的量子代数），重点关注代数结构和几何方面。这些代数是在20世纪80年代在解决理论量子物理问题的过程中发现的。它们是数学系统，在几何结构上模拟函数环，就像代数几何（研究由多项式方程定义的几何空间）一样，除了乘法是非交换的。量子代数的内部结构将被深入研究，特别是那些对应于经典类似物中的点和几何子空间的子结构，以及通过半经典极限（非交换性趋于零的过程）连接的量子代数和经典代数之间的关系。这些可交换的半经典极限代数带有相关量子代数的非交换性的剩余，记录在称为泊松括号的双线性形式中。这项工作也将追求密切相关的问题，泊松代数，以及经典量子之间的相互作用泊松半经典极限和它们的量子对应。这项研究的目的都是为了推进，并加强之间的联系，几个高度活跃的数学领域-量子群，泊松代数，非交换代数几何和集群理论。特别是，这项工作将提供工具，将见解从一个领域传递到其他领域。具体项目包括利用集群代数结构对大类量子和泊松代数对这些代数的理想理论。一个相关的努力将进入更可行的桥梁建设，在量子代数和它们的半经典极限之间来回转移信息。沿着这条线，调查员提出，素数和原始频谱的量子代数同胚的泊松素数和泊松原始频谱的半经典极限，他将投入主要精力进行确认。其他项目则针对经典量子相互作用。我们的目标是开发非交换版本的属性所享有的交换坐标环，其中突出的是他们的素谱链。本文的工作将探讨量子化坐标环具有链状性的猜想。这些努力的目标是阐明的整体拓扑结构的素数和原始谱的量子代数。这是为了通过辅助空间和在经典环境中不需要的映射来粘合这些谱的（已经固定的）经典片段。为这种粘合和辅助数据建立一个特定的、详细的格式，以及为量子代数和半经典极限的谱建立类似的、匹配的格式，将在这些谱之间产生明确的同胚，从而建立受约束的量子-泊松相互作用。