Representation Theory and Geometry in Monoidal Categories

幺半群范畴中的表示论和几何

• 批准号: 2401184
• 负责人: Daniel Nakano
• 金额: $ 25.53万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401184&HistoricalAwards=false
• 关键词: Representation; Theory; Geometry; Monoidal; Categories

The Principal Investigator (PI) will investigate the representation theory of various algebraic objects. A representation of an abstract algebraic object is a realization of the object via matrices of numbers. Often times, it is advantageous to view the entire collection of representations of an algebraic object as a structure known as a tensor category. Tensor categories consist of objects with additive and multiplicative operations like the integers or square matrices. Using the multiplicative operation, one can introduce the spectrum of the tensor category which is a geometric object (like a cone, sphere or torus). The PI will utilize the important connections between the algebraic and geometric properties of tensor categories to make advances in representation theory. The PI will continue to involve undergraduate and graduate students in these projects. He will continue to be an active member of the mathematical community by serving on national committees for the American Mathematical Society (AMS), and as an editor of a major mathematical journal.The PI will develop new methods to study monoidal triangular geometry. Several central problems will utilize the construction of homological primes in the general monoidal setting and the introduction of a representation theory for MTCs. This representation theory promises to yield new information about the Balmer spectrum of the MTC. In particular, the general MTC theory will be applied to study representations of Lie superalgebras. The PI will also explore new ideas to study representations of classical simple Lie superalgebras. This involves systematically studying various versions of Category O and the rational representations for the associated quasi-reductive supergroups. One of the main ideas entails the use of the detecting and BBW parabolic subgroups/subalgebras. Furthermore, the PI will study the orbit structure of the nilpotent cone and will construct resolutions of singularities for the orbit closures. The PI will study important questions involving representations of reductive algebraic groups. Key questions will focus on the understanding the structures of induced representations, and whether these modules admit p-filtrations. These questions are interrelated with the 30-year-old problem of realizing projective modules for the Frobenius kernels via tilting modules for the reductive algebraic group, and the structure of extensions between simple modules for the first Frobenius kernel.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

首席研究员（PI）将研究各种代数对象的表示论。抽象代数对象的表示是通过数字矩阵实现该对象。很多时候，将代数对象的整个表示集合视为称为张量类别的结构是有利的。张量类别由具有加法和乘法运算的对象组成，例如整数或方阵。使用乘法运算，我们可以引入作为几何对象（如圆锥体、球体或环面）的张量类别的谱。 PI 将利用张量类别的代数和几何性质之间的重要联系来推动表示论的发展。 PI 将继续让本科生和研究生参与这些项目。他将继续成为数学界的活跃成员，担任美国数学会 (AMS) 国家委员会的成员，并担任主要数学期刊的编辑。PI 将开发研究幺半群三角几何的新方法。几个核心问题将利用一般幺半群环境中同调素数的构造以及 MTC 表示理论的引入。这种表示理论有望产生有关 MTC 巴尔默谱的新信息。特别是，一般 MTC 理论将应用于研究李超代数的表示。 PI 还将探索新的想法来研究经典简单李超代数的表示。这涉及系统地研究 O 类的各种版本以及相关准还原超群的有理表示。主要思想之一是使用检测和 BBW 抛物线子群/子代数。此外，PI还将研究幂零锥的轨道结构，并构建轨道闭合的奇点分辨率。 PI 将研究涉及还原代数群表示的重要问题。关键问题将集中在理解诱导表征的结构，以及这些模块是否允许 p 过滤。这些问题与 30 年来通过还原代数群的倾斜模实现 Frobenius 核的射影模以及第一个 Frobenius 核的简单模之间的扩展结构相关。该奖项反映了 NSF 的法定使命和通过使用基金会的智力优点和更广泛的影响审查标准进行评估，该项目被认为值得支持。