Synopsis of the Conference
Hosted at the beautiful Tennessee Tech. University, the 2014 MAA-SE Conference brought together a wide variety of mathematics faculty and students to discuss mathematics and mathematics education. For more information on MAA-SE, please visit their website at http://sections.maa.org/southeastern/maase/.
My Presentation
During the graduate student talks, I was honored to present my current research on crystal basis theory of a particular finite Lie algebra, the special linear Lie algebra. The nature of this talk was to give a overview of what Crystal Basis Theory is, rather than a presentation of a new result. Below is the abstract for my talk and a PDF version of my presentation slides.
Abstract
On Crystal Basis Theory of sl(2,C)
The focus of this talk will be on the theory related to sl(2,C), the Lie algebra consisting of two by two complex matrices of trace zero. The quantum group for sl(2,C) is a deformation of its universal enveloping algebra, created by incorporating a quantum parameter q. Taking the limit as q approaches one for a representation of the quantum group, gives the corresponding representation for the universal enveloping algebra, which in turn corresponds to a representation of sl(2,C). Crystal bases for representations of the quantum group can be thought of as bases for these representations when q = 0. They parameterize representations for any q. While they may seem complicated at first, crystal bases exhibit numerous combinatorial patterns. As an example of this, the Tensor Product Rule will be briefly described along with preliminary, expository result from the finite version of sl(n,C).
The focus of this talk will be on the theory related to sl(2,C), the Lie algebra consisting of two by two complex matrices of trace zero. The quantum group for sl(2,C) is a deformation of its universal enveloping algebra, created by incorporating a quantum parameter q. Taking the limit as q approaches one for a representation of the quantum group, gives the corresponding representation for the universal enveloping algebra, which in turn corresponds to a representation of sl(2,C). Crystal bases for representations of the quantum group can be thought of as bases for these representations when q = 0. They parameterize representations for any q. While they may seem complicated at first, crystal bases exhibit numerous combinatorial patterns. As an example of this, the Tensor Product Rule will be briefly described along with preliminary, expository result from the finite version of sl(n,C).
Photos of Presentations
Below are photos of both mine and Thomas Cook's presentations given during the Graduate Student Presentation section of the conference. Mr. Cook presented on Complex Newton's Method, with a particular focus on the Newton map of complex cosine.