The Classification of Simple Lie Algebras in Maple

Author

D. Russell Sadler, Utah State University

Date of Award

2009

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

Committee Chair(s)

Mark Fels

Committee

Mark Fels

Committee

Ian Anderson

Committee

Zhi Q. Wang

Abstract

Lie algebras are invaluable tools in mathematics and physics as they enable us to study certain geometric objects such as Lie groups and differentiable manifolds. The computer algebra system Maple has several tools in its Lie Algebras package to work with Lie algebras and Lie groups. The purpose of this paper is to supplement the existing software with tools that are essential for the classification of simple Lie algebras over C.

In particular, we use a method to find a Cartan subalgebra of a Lie algebra in polynomial time. From the Cartan subalgebra we can compute the corresponding root system. This allows us to develop a command to compute the Cartan Matrix of a semisimple Lie algebra. From the Cartan Matrix we can construct the corresponding Dynkin diagram and determine the structure of the Lie algebra. We use the Cartan subalgebra and Cartan matrix to classify the simple Lie algebras over C.

We will also set out to define commands to initialize the classical Lie algebras of all dimensions in Maple. These commands will give us the tools needed to verify our results.

Recommended Citation

Sadler, D. Russell, "The Classification of Simple Lie Algebras in Maple" (2009). All Graduate Plan B and other Reports, Spring 1920 to Spring 2023. 1261.
https://digitalcommons.usu.edu/gradreports/1261