Excellence in Research: Cominuscule Quantum Schubert Cells and Faddeev-Reshetikhin-Takhtajian (FRT) Bialgebras

本课题研究了量子代数和非交换环理论中一系列相互关联的问题。量子群这个术语用来描述一大类代数，它们可以被认为是经典群和代数的变形。量子群的一些重要例子包括半单李代数的量子包络代数和代数群的量子化坐标环。量子群起源于物理学，但在拓扑学和统计力学等多个数学领域发挥了重要作用。构造代数群的量子化坐标环的原始路径是通过Faddeev、Reshetikhin和Takhtajian的一种通用双代数构造。由这种构造产生的双代数现在通常被称为泛frt双代数。量子舒伯特细胞代数是本研究项目中感兴趣的另一类代数。这些是量子化包络代数的负（或正）部分的子代数。近年来，它们在量子簇代数和编织对称代数中发挥了重要作用。我们将研究frt -双代数和量子舒伯特细胞代数之间的关系以及它们的环理论性质。该项目扩大了北卡罗来纳中央大学少数族裔学生参与研究的机会。在一些重要的情况下，量子舒伯特细胞代数可以看作是单幂李代数的量子化版本。量子舒伯特细胞最著名的例子是量子矩阵的代数。其他重要的例子包括量子对称矩阵和量子反对称矩阵的代数。在本研究计划中，我们将研究这类代数的环理论性质，例如它们的量子行列式、量子子式和量子法非子式。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。

This project investigates a series of interrelated problems in quantum algebras and noncommutative ring theory. The term quantum group is used to describe a large class of algebras which can be thought of as deformations of classical groups and algebras. Some important examples of quantum groups include the quantum enveloping algebras of semisimple Lie algebras and the quantized coordinate rings of algebraic groups. Quantum groups originated in physics, but have since played important roles in multiple areas of mathematics, such as topology and statistical mechanics. The original path to constructing the quantized coordinate ring of an algebraic group is through a universal bialgebra construction due to Faddeev, Reshetikhin, and Takhtajian. The bialgebras arising from this construction are now commonly referred to as universal FRT-bialgebras. Quantum Schubert cell algebras are another family of algebras of interest in this research project. These are certain subalgebras of the negative (or positive) part of a quantized enveloping algebra. They have played important roles recently in the context of quantum cluster algebras and braided symmetric algebras. We will study relationships between FRT-bialgebras and quantum Schubert cell algebras as well as their ring-theoretical properties. This project expands opportunities for student research involvement drawn from the predominantly underrepresented minority student population at North Carolina Central University.In several important cases, quantum Schubert cell algebras may be regarded as quantized versions of unipotent Lie algebras. The most famous examples of quantum Schubert cells are the algebras of quantum matrices. Other important examples include the algebras of quantum symmetric matrices and quantum anti-symmetric matrices. In this research project we will study ring-theoretic properties of these types of algebras, such as their quantum determinants, quantum minors, and quantum Pfaffians.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.