Noncommutative Algebras and Related Categorical Structures

科学和工程中的许多模型都会导致关于交换变量方程解的数学问题。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交换变量的数学问题。非交换代数是研究这些结构的数学领域之一。由该奖项资助的研究项目利用称为泊松几何的几何分支，研究交换和非交换设置之间的相互关系。另外三种方法用于研究非交换设置：一种基于对象复杂的内部变换的组合方法，称为簇突变；一种研究被称为Calabi-Yau范畴的内在定义结构的代数方法；以及一种非交换几何方法利用量子版本的对称空间。同时使用这四种方法对非交换对象的性质和对称性进行了详细的研究。此外，非交换对象显示出各种形式的刚性。它用于解决代数、几何、组合子和动力系统中的问题，这些问题以前在没有任何参考非交换设置的情况下提出。这些研究活动将作为培养研究生和本科生以及指导数学博士后的基础。该奖项资助的研究项目主要研究非交换代数、量子对称空间和非交换投影代数几何中的问题，以及这些问题与泊松几何、组合学、三角范畴和可积系统的关系。一方面，该计划旨在使用后一个领域的方法来描述在单位根处的量子簇代数的结构和表示，Nichols代数的Drinfeld double，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，先前提出的问题在后一个领域被转换为问题的非交换代数及其表示范畴，然后在该设置解决。本程序的一个方向是在Nichols代数的Drinfeld双元的对称子代数上构造泛k矩阵，并以此来研究Nichols代数的环论性质。另一个方向是利用泊松阶和非交换判别式对对角型Nichols代数的不可约表示进行分类。第三个方向是利用泊松几何和Cayley-Hamilton代数研究单位根处量子簇代数的有限维表示的一般设置。另外三个方向研究了由高维椭圆代数建模的非交换射影空间的几何，通过范畴c向量和动力系统的2-Calabi-Yau范畴的结构，以及李群理论中量子化变异坐标环规范形式上的积分量子聚类代数结构的构造。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.