#### Year

2020

#### Season

Spring

#### Paper Type

Master's Thesis

#### College

College of Arts and Sciences

#### Degree Name

Master of Science in Mathematical Sciences (MS)

#### Department

Mathematics & Statistics

#### NACO controlled Corporate Body

University of North Florida. Department of Mathematics and Statistics

#### First Advisor

Dr. Jae-Ho Lee

#### Department Chair

Dr. Richard Patterson

#### Abstract

We study the hypercube and the associated subconstituent algebra. Let Q_D denote the hypercube with dimension D and let X denote the vertex set of Q_D. Fix a vertex x in X. We denote by A the adjacency matrix of Q_D and by A* = A*(x) the diagonal matrix with yy-entry equal to D − 2i, where i is the distance between x and y. The subconstitutent algebra T = T(x) of Q_D with respect to x is generated by A and A* . We show that A 2A* − 2AA*A + A*A 2 = 4A* A*2A − 2A*AA* + AA*2 = 4A. Using these relations, we show that there exists a surjective C-algebra homomorphism from the universal enveloping algebra of the Lie algebra sl2(C) to T.

#### Suggested Citation

Billet, Jared B., "The Subconstituent Algebra of a Hypercube" (2020). *UNF Graduate Theses and Dissertations*. 945.

https://digitalcommons.unf.edu/etd/945