Noncommutative Algebraic Geometry and Noncommutative Invariant Theory

非交换代数几何和非交换不变理论

• 批准号: 1401207
• 负责人: Chelsea Walton
• 金额: $ 14.84万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401207&HistoricalAwards=false
• 关键词: Noncommutative; Algebraic; Geometry; Invariant; Theory

This research project concerns noncommutative algebra with connections to both invariant theory and algebraic geometry. An algebra is a fundamental object throughout mathematics. One use of algebras is to encode information about geometric structures. Many algebras arising in mathematics and physics are noncommutative, i.e., the product of two elements depends on the order in which elements are multiplied. The PI will investigate noncommutative algebras that are related to noncommutative geometry and noncommutative invariant theory.The research of the principal investigator lies in two subfields of noncommutative algebra entitled Noncommutative Projective Algebraic Geometry (NCPAG) and Noncommutative Invariant Theory (NCIT). The first area was launched in the 1980s to examine algebras, especially the three-dimensional Sklyanin algebras Skly3, whose ring-theoretic behavior could not be determined using purely algebraic techniques. The PI plans to continue using techniques of NCPAG to establish results on Skly3 and on algebras arising in physics and Lie theory, such as the Virasoro algebra and other related algebras. The goal of the second area of research, NCIT, is to extend results in classical invariant theory to a noncommutative setting. To do so, we replace an action of a group on a commutative polynomial ring by an action of a Hopf algebra (or quantum group) on a noncommutative regular algebra that shares homological properties with its commutative counterpart. Although it is difficult for a Hopf algebra to act on an algebra, the PI has works in progress on actions of Hopf algebras on commutative domains, on their quantizations, and on algebras arising from NCPAG. The PI also intends to continue to contribute results on algebras arising from these actions, such as invariant subrings and smash product algebras. This will lead to new examples of algebras whose representation theory merit further investigation.

该研究项目涉及与不变理论和代数几何相关的非交换代数。代数是整个数学的基本对象。代数的用途之一是对有关几何结构的信息进行编码。数学和物理学中出现的许多代数都是不可交换的，即两个元素的乘积取决于元素相乘的顺序。 PI将研究与非交换几何和非交换不变量理论相关的非交换代数。主要研究人员的研究集中在非交换代数的两个子领域，即非交换射影代数几何（NCPAG）和非交换不变量理论（NCIT）。第一个领域于 20 世纪 80 年代启动，旨在研究代数，特别是三维 Sklyanin 代数 Skly3，其环理论行为无法使用纯代数技术来确定。 PI 计划继续使用 NCPAG 的技术在 Skly3 以及物理学和李理论中出现的代数（例如 Virasoro 代数和其他相关代数）上建立结果。 第二个研究领域 NCIT 的目标是将经典不变量理论的结果扩展到非交换环境。为此，我们将交换多项式环上的群的作用替换为非交换正则代数上的 Hopf 代数（或量子群）的作用，该代数与其交换对应代数具有同调性质。尽管 Hopf 代数很难作用于代数，但 PI 正在研究 Hopf 代数对交换域、量子化以及 NCPAG 产生的代数的作用。 PI 还打算继续贡献这些行动所产生的代数成果，例如不变子环和粉碎积代数。这将产生新的代数例子，其表示理论值得进一步研究。