Geometric and its applications in algebra

Here we introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to define Clifford algebras with scalars in arbitrary rings and provides new suggestions for an infinite-dimensional approach. Furthermore, we give a quick review of classic results regarding geometric algebras, such as their classification, the connection to orthogonal and Spin groups, and their representation theory. A number of lower-dimensional Patterns are worked out in a systematic way using so called norm functions, while general applications of representation theory include normed division algebras and vector fields on spheres. We also consider Patterns in relativistic physics, where reformulations in terms of geometric algebra give rise to both computational and conceptual simplifications.

Keywords:

Geometric , Spin groups , algebra , conceptual simplifications