Operator algebras are collections of linear functions on geometric spaces of vectors called Hilbert spaces. Operator algebras represent collections of physical observables in quantum mechanics, and have also found application to many other mathematical fields. In the last 50 years, the study of operator algebras which are "non-selfadjoint", such as algebras of triangular matrices, has been especially fruitful. Of special interest in modern research-level mathematics are non-selfadjoint operator algebras which are "semi-Dirichlet".

In this project, a MacEwan undergraduate research assistant will instigate a theoretical and/or computer classification of those finite-dimensional operator algebras which are semi-Dirichlet. In the most basic cases, such algebras can be classified using directed graphs. A celebrated theorem of Burnside describes the structure of any finite-dimensional algebra, and a classification of the semi-Dirichlet property will combine graph theory and Burnside's Theorem.

By the end of this project, the student will produce an article suitable for publication in an undergraduate or research-level journal. This is new mathematical research carried out by an undergraduate researcher at MacEwan University, supervised by Drs. Adam Humeniuk and Chris Ramsey.

In this first phase, the student will become familiar with the background material, produce many examples of semi-Dirichlet and non-semi-Dirichlet algebras, and begin writing up in LaTex the initial material. In particular, this will focus on the directed graph special case.

We will meet weekly throughout this project in-person or online, providing background information to the student and instruction on writing in LaTex (the mathematical typesetting language). Additionally, we will focus on skills development such as project management, searching for resources, programming, and publishing.