Noncommutative Algebras and Monoidal Triangulated Categories

科学和工程中的许多模型都是基于涉及通勤变量的数学设置。然而，从量子力学开始，出现了大量的模型，这些模型导致了涉及不再往返的变量的重要数学问题。非对易代数是研究这些结构的主要数学领域之一。这个项目解决了关于非交换对象的性质和对称性的关键问题，以及相应的非交换代数的表示。表示论的后一个领域通过所有可能的方法来研究非交换代数，以矩阵的形式来表示它们。使用了三种主要的方法：(1)泊松几何--从交换对象到非交换对象的变形而产生的几何；(2)簇代数--一种基于复杂的对象内部变换的组合方法，称为簇突变；(3)么半三角形范畴--由同时考虑一个代数的所有表示而产生的一般抽象代数结构。这些研究活动将被用作培训研究生和本科生以及指导数学博士后的基础。更详细地说，在这个项目中，PI将研究量子对称空间、Nichols代数、单面三角范畴的结构，以及有限维代数的支撑理论，更广泛地说，有限张量范畴。主要的研究方向如下：(1)量子簇代数的单位根在两个相互关联的方案中被研究：它们的判别理想的描述和不可约表示的分类。这将以泊松序理论和Cayley-Hamilton代数为基础。(2)在量子旗簇、量子Bott-Samelson簇和量子对称空间上构造量子簇代数结构。我们将通过第一部分中的技巧研究量子单位根对应的不可约表示和判别理想。利用星积和泊松序发展量子对称对和单位根处的量子超群的表示理论。(3)建立有限张量范畴的稳定范畴的非对易Balmer谱的分类方法。它们将被用来描述有限维Hopf代数的上同调支撑，更广泛地说，有限张量范畴。这个奖项反映了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。

Many models in the sciences and engineering are based on mathematical settings that involve commuting variables. However, starting with quantum mechanics, a great number of models emerged that led to important mathematical problems involving variables that no longer commute. Noncommutative Algebra is one of the major areas of mathematics that studies those structures. This project addresses key problems about the properties and symmetries of noncommutative objects, as well as the representations of the corresponding noncommutative algebras. The latter area of representations theory investigates noncommutative algebras through all possible ways to present them in terms of matrices. Three major approaches are used: (1) Poisson geometry--geometry arising from the deformation of commutative objects to noncommutative ones, (2) cluster algebras--a combinatorial approach based on intricate internal transformations of the objects, called cluster mutations and (3) monoidal triangulated categories--general abstract algebraic structures arising from considering all representations of an algebra simultaneously. These research activities will be used as the foundation for the training of graduate and undergraduate students and for mentoring of mathematics postdocs. In more detail, in this project the PI will investigate the structure of quantum symmetric spaces, Nichols algebras, monoidal triangulated categories, as well as support theories for finite dimensional algebras and, more generally, finite tensor categories. The following three broad directions will be pursued: (1) Root of unity quantum cluster algebras will be investigated in two interrelated plans: description of their discriminant ideals and classification of irreducible representations. This will be based on the theory of Poisson orders and Cayley-Hamilton algebras. (2) Quantum cluster algebra structures will be constructed on quantum flag varieties, quantum Bott-Samelson varieties and quantum symmetric spaces. The irreducible representations and discriminant ideals of their root of unity quantum counterparts will be studied through the techniques developed in part 1. The representation theory of quantum symmetric pairs and quantum supergroups at roots of unity will be developed using star products and Poisson orders. (3) Methods for the classification of the noncommutative Balmer spectra of the stable categories of finite tensor categories will be developed. They will be used for the descriptions of the cohomological supports of finite dimensional Hopf algebras, and more generally, finite tensor categories.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.