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

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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.