Abstract

Geometric algebra is a mathematical field which roots go back to Leibniz’s “Geometrische Analyse” and which is concerned with algebraizing geometry in such a way as to keep the prospects of synthetic and analytic reasoning.

It was developed mainly in the 19th century and its foundations foremost by Grassmann with his theory of linear extension. Many other types of algebras like various quaternionic algebras and more generally Clifford algebras stem from geometric algebra, too. Although many of those particular algebras are still prominent, the field of geometric algebra itself has been nearly completely forgotten in the 20th century.

My thesis is concerned with the foundations of geometric algebras of affine space. The central question is thereby “What is a geometric algebra of an affine space?”.

One of the main goals is thus to understand which categories of algebras are natural candidates for geometric algebras of an affine space to live in.