Excellence in Research: Super Representation Theory, Quantum Spaces, and Algebra Maps

Algebras are objects which reveal symmetries of the physical world and strengthen understanding of physics, chemistry, quantum information, biology, and their mathematical connections. These symmetries are organized via a mathematical framework called representation theory. Finite-dimensional representation theory involves the realization of an algebra as arrays of numbers (matrices); infinite-dimensional representation theory involves the realization of an algebra as functions, including differentiation from calculus. This project focuses on the infinite-dimensional representations of superalgebras as motivated by supersymmetry of particle physics. The Principal Investigator (PI) expects three main results from the project: (1) creation of new infinite-dimensional super representations; (2) simplification of super representations into their constituent parts; and, (3) determination of formulas to allow for computation and increased applicability of representation theory to scientific phenomena. The award also supports undergraduate research trainees and a doctoral student supervised by the PI in solving one or more of the research problems of the project. In more detail, the project has three main technical parts. The first part is that the PI will establish superalgebra homomorphisms from the universal enveloping algebra of orthosymplectic Lie superalgebras to tensor products of Clifford-Weyl superalgebras. The second part relies on the PI building upon previous collaborative work to determine generators and relations of reduction superalgebras. In the third part, the PI will decompose the representations from part one through the application of reduction superalgebra techniques. The expected results are novel and concrete examples of reduction superalgebras, which are of interest in the study of integrable systems and as symmetry objects of quantum spaces. The results will advance progress in finding bases with formulas for the action of orthosymplectic Lie superalgebras on simple infinite-dimensional representations in an explicit contribution to an overarching problem of super representation theory. 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.