Noncommutative Algebras and Related Categorical Structures

非交换代数和相关分类结构

基本信息

批准号：
1901830
负责人：
Milen Yakimov
金额：
$ 34.51万
依托单位：
Louisiana State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2021-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901830&HistoricalAwards=false
关键词：
Noncommutative Algebras Related Categorical Structures

项目摘要

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.

科学和工程中的许多模型导致有关通勤变量方程解决方案的数学问题。但是，从量子力学开始，出现了许多模型，这些模型导致了数学问题，涉及不再通勤的变量。非共同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使用称为Poisson几何形状的分支，调查了交换和非交通设置之间的相互关系。其他三种方法用于研究非共同设置：基于复杂的对象的内部变换（称为群集突变）的组合方法；一种代数方法，研究了本质上定义的结构称为calabi-yau类别；以及使用对称空间的量子版本的非共同几何方法。这四种方法同时用于对非交通对象的性质和对称性进行详细研究。此外，表明非共同对象表现出各种形式的刚性。这用于在代数，几何形状，组合器和动力学系统中解决问题，这些问题先前构成了无通用设置。 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.一方面，该计划旨在使用后者区域的方法来描述统一根部的量子群集代数的结构和表示，即nichols代数的Drinfeld双打，即量子对称对理论中出现的代数，以及描述非共同投影空间的代数。在相反的方向上，以前在后一个区域中提出的问题被转换为非共同代数及其表示类别的问题，然后在该环境中解决。该程序的方向之一是在尼科尔（Nichols）代数的德林菲尔德双打对称亚代代代代代代代代代代代代代数上的构建通用k-矩阵，并使用它来研究尼科尔斯代数的环理论特性。另一个方向旨在使用泊松命令和非交通性判别因子对对角类型的尼科尔（Nichols）代数的不可减至表示。第三个方向开发了研究使用泊松几何形状和cayley-hamilton代数的量子群集代数的有限维表示的一般设置。三个其他方向研究了由较高维椭圆形的代数建模的非共同投影空间的几何形状，通过分类c-媒介和动力学系统的2-卡拉比YAU类别的结构，以及整体量子集群代数的结构认为值得通过基金会的智力优点和更广泛影响的评论标准来评估值得支持。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Integral quantum cluster structures

DOI：
10.1215/00127094-2020-0061
发表时间：
2020-03
期刊：
Duke Mathematical Journal
影响因子：
2.5
作者：
K. Goodearl;M. Yakimov
通讯作者：
K. Goodearl;M. Yakimov

Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

双变量连续 q-Hermite 多项式和变形量子 Serre 关系

DOI：
10.1142/s0219498821400168
发表时间：
2021
期刊：
Journal of Algebra and Its Applications
影响因子：
0.8
作者：
Riley Casper, W.;Kolb, Stefan;Yakimov, Milen
通讯作者：
Yakimov, Milen

Poisson orders on large quantum groups

大量子群的泊松阶

DOI：
10.1016/j.aim.2023.109134
发表时间：
2023
期刊：
Advances in Mathematics
影响因子：
1.7
作者：
Andruskiewitsch, Nicolás;Angiono, Iván;Yakimov, Milen
通讯作者：
Yakimov, Milen

Reflective prolate-spheroidal operators and the KP/KdV equations

DOI：
10.1073/pnas.1906098116
发表时间：
2019-09-10
期刊：
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
影响因子：
11.1
作者：
Casper, W. Riley;Grunbaum, F. Alberto;Zurrian, Ignacio
通讯作者：
Zurrian, Ignacio

Integral operators, bispectrality and growth of Fourier algebras

积分算子、双谱性和傅里叶代数的增长