Lie Algebras and Algebra of Incidence Using the Null Cone and Affine Plane

Abstract

In this chapter we give the fundamentals of Lie algebra and the algebra of incidence using the computational frameworks of the null cone and the n-dimensional affine plane. Using Lie algebra within this computational framework has the advantage that it is easily accessible to the reader because there is a direct translation of the familiar matrix representations to representations using bivectors from geometric algebra. Generally speaking, Lie group theory is the appropriate tool for the study and analysis of the action of a group on a manifold. Since we are interested in purely geometric computations, the use of geometric algebra is appropriate for carrying out computations in a coordinate-free manner by using a bivector representation of the most important Lie algebras. We will represent Lie operators using bivectors for the computation of a variety of invariants. This chapter benefits of work done in colaboration with Sobczyk [11, 100].