Noncommutative Algebras and Related Categorical Structures

科学和工程中的许多模型导致了关于交换变量方程的解的数学问题。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往返的变量的数学问题。非对易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使用称为泊松几何的几何学分支来研究对易和非对易环境之间的相互关系。其他三种方法被用来研究非对易环境：一种基于复杂的对象内部变换的组合方法，称为簇突变；一种代数方法，研究被称为Calabi-Yau范畴的内在定义结构；以及一种使用对称空间的量子版本的非对易几何方法。这四种方法同时被用来详细地研究非对易对象的性质和对称性。此外，非对易对象显示出各种形式的刚性。它被用来解决代数、几何、组合子和动力系统中的问题，这些问题以前是在没有任何非对易设置的情况下提出的。这些研究活动将被用作培养研究生和本科生以及指导数学博士后的基础。在此基础上资助的研究项目研究了非交换代数、量子对称空间和非交换射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，该程序旨在使用后一领域的方法来描述单位根处的量子簇代数、Nichols代数的Drinfeld对偶、量子对称对理论中出现的代数以及描述非对易射影空间的代数的结构和表示。在相反的方向上，将后一领域中先前提出的问题转化为非对易代数及其表示范畴的问题，然后在该环境下求解。这个程序的一个方向是构造Nichols代数的Drinfeld对称子代数上的泛K-矩阵，并用它来研究Nichols代数的环论性质。另一个方向是利用Poisson序和非交换判别式对对角型Nichols代数的不可约表示进行分类。第三个方向发展了利用泊松几何和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-向量和动力系统构造2-Calabi-Yau范畴，以及在李群理论中各种量子化坐标环的典范形式上构造积分量子簇代数结构。这一奖项反映了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 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.