On Tuesday, February 5 (my birthday!), I gave a talk titled “A diagrammatic representation of the Temperley-Lieb algebra” in the NAU Department of Mathematics and Statistics Colloquium. Here is the abstract.
One aspect of my research involves trying to prove that certain associative algebras can be faithfully represented using “diagrams.” These diagrammatic representations are not only nice to look at, but they also help us recognize things about the original algebra that we may not otherwise have noticed. In this talk, we will introduce the diagram calculus for the Temperley-Lieb algebra of type $A$. This algebra, invented by Temperley and Lieb in 1971, is a certain finite dimensional associative algebra that arose in the context of statistical mechanics in physics. We will show that this algebra has dimension equal to the nth Catalan number and discuss its relationship to the symmetric group. If time permits, we will also briefly discuss the diagrammatic representation of the Temperley-Lieb algebra of type affine $C$.
And here are the slides.
Despite the fact that this was a 50-minute talk, it was intended to be an overview of one aspect of a long and complex story. The subject matter is intimately related to my PhD thesis, as well as a series of papers that I have written.
In addition, there is (at least) a part III that goes with the last two papers that is in progress.
Mathematics & Teaching
Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.
The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.
The source code is on GitHub.
Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.