In the 1920s, Heisenberg developed the Fundamental Equation of Quantum Mechanics to represent the relationship between position and momentum of a particle. To study the equation, they started learning Generalized Weyl Algebras (GWAs). GWAs appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariant theory of certain GWAs, called quantum GWAs. Invariant theory is the study of algebraic or geometric properties of an object that are preserved under symmetry. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We also study properties of the fixed rings, including the global dimension, rigidity, and simplicity. I started working on my research by using examples and this led to a proof of a more general theorem. Working on this research with Dr. Gaddis, I have gained a lot of math problem-solving skills and a deep-rooted understanding of the mathematical field of my interest throughout those two months.
Author: Phuong Ho
Advisor: Jason Gaddis, Department of Mathematics
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