Noncommutative Algebras and Related Categorical Structures

科学和工程中的许多模型都导致了关于交换变量方程解的数学问题。然而，从量子力学开始，大量的模型出现，导致涉及不再交换的变量的数学问题。非交换代数是研究这些结构的数学领域之一。该奖项资助的研究项目研究交换和非交换设置之间的相互关系，使用几何学的分支称为泊松几何。其他三种方法被用来研究非交换设置：组合方法的基础上复杂的内部变换的对象，称为集群突变;代数方法，调查内在定义的结构称为卡-丘范畴;和非交换几何方法使用量子版本的对称空间。这四种方法同时被用来进行详细的研究的性质和对称性的非交换对象。此外，非对易对象表现出各种形式的刚性。这是用来解决问题的代数，几何，组合子，和动力系统，以前提出没有任何参考的非交换设置。这些研究活动将被用来作为培养研究生和本科生的基础，并指导数学博士后。在此awaard资助的研究项目调查的问题，在非交换代数，量子对称空间，和非交换射影代数几何和这些问题的关系，泊松几何，组合数学，三角范畴，和可积系统。一方面，该计划旨在使用后一领域的方法来描述量子簇代数在单位根的结构和表示，Nichols代数的Drinfeld双，量子对称对理论中出现的代数，以及描述非交换射影空间的代数。在相反的方向，以前提出的问题，在后者的领域被转换为问题的非交换代数及其表示类别，然后解决该设置。该程序的一个方向是构造Nichols代数的Drinfeld双的对称子代数上的泛K-矩阵，并利用它来研究Nichols代数的环论性质。另一个方向的目的是分类的不可约表示的尼科尔斯代数的对角型使用泊松秩序和非交换判别式。第三个方向开发了一个通用的设置为研究有限维表示的量子集群代数在根单位使用泊松几何和凯莱-汉密尔顿代数。三个额外的方向调查的几何非交换射影空间建模的高维椭圆代数，结构的2-Calabi-Yau范畴通过范畴C-向量和动力系统，以及在李群理论中的簇的量子化坐标环的标准形上构造积分量子簇代数结构。该奖项反映了NSF的法定使命，并被认为值得支持通过使用基金会的知识价值和更广泛的影响审查标准进行评估。

Many models in the sciences and engineering lead to mathematical problems about solutions of equations in commuting variables. However, starting with quantum mechanics, a great number of models emerged that led to mathematical problems involving variables that no longer commute. Noncommutative Algebra is one of the areas of mathematics that studies those structures. The research projects that are funded by this award investigate the interrelations between the commutative and noncommutative settings, using a branch of geometry called Poisson geometry. Three other approaches are used to study the noncommutative setting: a combinatorial approach based on intricate internal transformations of the objects, called cluster mutations; an algebraic approach that investigates intrinsically defined structures called Calabi-Yau categories; and a noncommutative geometric approach using quantum versions of symmetric spaces. The four approaches are simultaneously used to carry out a detailed study of the properties and symmetries of noncommutative objects. Further, the noncommutative objects are shown to exhibit various forms of rigidity. This is used to settle problems in algebra, geometry, combinators, and dynamical systems that were previously posed without any reference to the noncommutative setting. These research activities will be used as the foundation for the training of graduate and undergraduate students and for mentoring mathematics postdocs.The research projects funded under this awaard investigate problems in noncommutative algebra, quantum symmetric spaces, and noncommutative projective algebraic geometry and the relations of these problems to Poisson geometry, combinatorics, triangulated categories, and integrable systems. On the one hand, the program aims at using methods from the latter areas to describe the structure and representations of quantum cluster algebras at roots of unity, the Drinfeld doubles of Nichols algebras, the algebras that appear in the theory of quantum symmetric pairs, and the algebras that describe noncommutative projective spaces. In the opposite direction, previously posed problems in the latter areas are converted to problems for noncommutative algebras and their representation categories, and are then resolved within that setting. One of the directions of this program is the construction of universal K-matrices on the symmetric subalgebras of the Drinfeld doubles of Nichols algebras, and using this to study the ring theoretic properties of Nichols algebras. Another direction aims at the classification of irreducible representations of Nichols algebras of diagonal type using Poisson orders and noncommutative discriminants. A third direction develops a general setting for the study of finite dimensional representations of quantum cluster algebras at root of unity using Poisson geometry and Cayley-Hamilton algebras. Three additional directions investigate the geometry of noncommutative projective spaces modeled by higher dimensional elliptic algebras, the structure of 2-Calabi-Yau categories via categorical C-vectors and dynamical systems, and the construction of integral quantum cluster algebra structures on the canonical forms of quantized coordinate rings of varieties in theory of Lie groups.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.