Introduction into Geometry over Division Ring

TL;DR

This paper explores the geometry of affine and Euclidean spaces over division rings, extending representation theory of F-algebras and examining structures like manifolds with affine connections.

Contribution

It introduces the concept of towers of representations of F_i-algebras and applies these ideas to the geometry of affine and Euclidean spaces over division rings.

Findings

Representation theory of F-algebras extended to geometric structures.

Analysis of affine space and manifolds with affine connection over division rings.

Application of curvilinear coordinates to these geometric frameworks.

Abstract

Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the notion of tower of representations of F_i-algebras, i=1, ..., n, as the set of coordinated representations of F_i-algebras. I explore the geometry of affine space as example of tower of representations. Exploration of curvilinear coordinates allows us to see how look at the main structures of the manifold with affine connection. I explore Euclidean space over division ring.