Monoidal Triangular Categories: Representation Theory, Cohomology, and Geometry

表示论已成为现代数学的中心领域，与组合学、代数几何、拓扑学和数论有着密切的联系。此外，它在化学和物理中也有许多应用。表示是复杂的代数对象(如群、环、李代数和李超代数)到一组数字(矩阵)的映射。这些通过矩阵实现的编码可以产生对这些复杂的代数对象的深刻见解的重要数据。近年来，一种有用的方法是理解对象的整个表示集合。一个代数对象的表示通常形成一个张量三角范畴。同调代数中的技巧可以用来在张量三角范畴和几何对象之间架起桥梁。揭示这种隐藏的几何图形通常会带来对代数对象及其表示的新的见解。该项目包括为研究生和博士后在代数和表示理论方面的研究和培训机会。在这个项目中，PI将开发新的方法来理解一元张量范畴中的张量结构。这将需要单面三角形几何的发展和巴尔默光谱的显式计算。PI还将引入新的几何和拓扑方法，以提供代数/有限群、李超代数、量子群和Frobenius核的上同调的具体计算。上同调理论的发展是解决30年来处理与代数群表示的滤子有关的倾斜模的问题的重要工具。PI将通过应用超几何来构造表示和计算更高的层上同调来研究李超代数的表示。PI将开发一种新的测谎论，它需要使用检测和抛物子代数，此外，还需要使用幂零锥来制定几何设置，以便计算不可约表示的特征。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。

Representation theory has emerged as a central area of modern mathematics with connections to combinatorics, algebraic geometry, topology and number theory. In addition, it has many applications to chemistry and physics. Representations are mappings of complicated algebraic objects (such as groups, rings, Lie algebras, and Lie superalgebras) to an array of numbers (matrices). These realizations by matrices encode important data that can yield deep insights into these complicated algebraic objects. In recent years, a useful approach has been to understand the entire collection of representations of an object. Representations for a certain algebraic object often form a tensor triangulated category. Techniques from homological algebra can be used to build bridges between tensor triangulated categories and geometric objects. Uncovering this hidden geometry often leads to new insights about the algebraic object and its representations. This project includes research and training opportunities for graduate students and postdoctoral fellows in algebra and representation theory. In this project the PI will develop new methods to understand tensor structures in monoidal tensor categories. This will entail the development of monoidal triangular geometry and explicit computations of Balmer spectra. The PI will also introduce new geometric and topological methods to provide concrete calculations of cohomology for algebraic/finite groups, Lie superalgebras, quantum groups, and Frobenius kernels. The development of the cohomology theory was an important tool to resolving 30 year old problems that deal with tilting modules in connection with filtrations of representations of algebraic groups. The PI will study representations of Lie superalgebras via the application of super geometry to construct representations and to compute higher sheaf cohomology. The PI will develop a new Lie theory that entails the use of detecting and parabolic subalgebras, in addition to, a nilpotent cone to formulate a geometric setting in order to compute characters for irreducible representations.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.