Excellence in Research: Quantum nilpotent algebras and essential sets

Quantum algebras are mathematical objects that arose from problems in quantum physics. In many instances, they are constructed by deforming the structure of classical algebras, which serve as mathematical models in classical physics. The quantum structure is usually noncommutative (meaning the variables do not commute with each other), whereas the classical structure is commutative (the variables commute). Quantum algebras have since played important roles in several areas of mathematics, including topology, statistical mechanics, and noncommutative algebraic geometry. This research project studies key questions involving the algebraic structures and symmetries of various classes of these algebras. The work also pursues related questions about Fulton's essential sets. These are combinatorial objects that originated in the early 1990's from problems in Schubert calculus, and have since appeared in various mathematical contexts, including Lie theory and Coxeter group theory. The research aims to advance, and to strengthen connections among, these interrelated areas. This project expands opportunities for student research involvement at North Carolina Central University. Specifically, this Principal Investigator will study relationships between quantum nilpotent algebras and Fulton's essential sets, and explore applications to representation theory and other structural properties, particularly automorphisms, of these algebras. The major research goals of this project include: (1) classifying/enumerating the quantum Schubert cell algebras that have only trivial unipotent automorphisms, and those having only graded automorphisms, (2) characterizing nilpotency indices for quantum nilpotent algebras explicitly in terms of essential sets, and (3) determining the automorphism groups of quantized nilradicals. The anticipated outcomes of this project are to develop a more complete theory of essential sets in cases outside of type A, and to develop computational methods that would be applicable to classifying certain types of morphisms of quantum nilpotent algebras. 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.