Applications of Lie Algebras and the Algebra of Incidence

摘要

We present the fundamentals of Lie algebra and the algebra of incidence in the n-dimensional affine plane. The difference between our approach and previous contributions, [ 5, 4, 2] is twofold. First, our approach is easily accessible to the reader because there is a direct translation of the familiar matrix representations to our representation using bivectors from the appropriate geometric algebra. Second, our "hands on" approach provides examples from robotics and image analysis so that the reader can become familiar with the computational aspects of the problems involved. This chapter is to some extent complimentary to the above mentioned references. Lie group theory is the appropriate tool for the study and analysis of the action of a group on a manifold. Geometric algebra makes it possible to carry out computations in a coordinate-free manner by using a bivector representation of the most important Lie algebras [ 5]. Using the bivector representation of a Lie operator, we can easily compute a variety of invariants useful in robotics and image analysis. In our study of rigid motion in the n-dimensional affine plane, we use both the structure of the Lie algebra alongside the operations of meet and join from incidence algebra.