Representation Theory and Geometry in Monoidal Categories

首席调查员(PI)将研究各种代数对象的表示理论。抽象代数对象的表示是对象通过数字矩阵的实现。通常，将代数对象的整个表示集合视为称为张量范畴的结构是有利的。张量范畴由具有加法和乘法运算的对象组成，如整数或方阵。利用乘法运算，可以引入几何对象(如锥体、球体或环面)的张量范畴的谱。PI将利用张量范畴的代数和几何性质之间的重要联系来推动表示理论的发展。PI将继续让本科生和研究生参与这些项目。他将继续担任美国数学学会(AMS)国家委员会的活跃成员，并担任一本主要数学期刊的编辑。PI将开发研究一元三角几何的新方法。几个中心问题将利用一般么半群环境中的同调素数的构造和MTC的表示理论的引入。这一表象理论有望提供有关MTC的巴尔默光谱的新信息。特别是，一般的MTC理论将被应用于研究李超代数的表示。PI还将探索新的思想来研究经典单李超代数的表示。这涉及到系统地研究范畴O的各种形式以及相关拟约化超群的有理表示。其中一个主要思想是使用检测和BBW抛物子群/子代数。此外，PI将研究幂零锥的轨道结构，并将构造轨道闭合的奇点解析。PI将研究涉及约化代数群表示的重要问题。关键问题将集中在对诱导表示的结构的理解上，以及这些模块是否允许p-过滤。这些问题与30年来的问题相互关联，即通过还原代数群的倾斜模来实现Frobenius核的投影模，以及第一个Frobenius核的简单模之间的扩展结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.