The Classification of Simple Lie Algebras in Maple

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

Computing Cartan subalgebras of Lie algebras

We consider the algorithmic problem of computing Cartan subalgebras in Lie algebras over finite fields and algebraic number fields. We present a deterministic polynomial time algorithm for the case when the ground fieldk is sufficiently large. Our method is based on a solution of a linear algebra problem: the task of finding a locally regular element in a subspace of linear transformations. Also, we give a polynomial time algorithm for restricted Lie algebras over arbitrary finite fields. Both methods require an auxiliary procedure for finding non-nilpotent elements in subalgebras. This problem is also treated. Computational experiences are discussed at the end of the paper