Direct Construction of Grassmann, Clifford and Geometric Algebras

TL;DR

This paper presents a straightforward, rigorous method for constructing Grassmann, Clifford, and Geometric Algebras that accommodates degenerate forms, infinite dimensions, and various fields, with a focus on universality and coordinate-free characterization.

Contribution

It introduces a simple, universal construction method for these algebras that handles degenerate forms, infinite dimensions, and is characterized in a coordinate-free manner.

Findings

Construction method is rigorous and simple

Handles degenerate bilinear forms and infinite dimensions

Provides a clear, foundational framework

Abstract

This is a simple way rigorously to construct Grassmann, Clifford and Geometric Algebras, allowing degenerate bilinear forms, infinite dimension, using fields or certain modules (characteristic 2 with limitation) - and characterize the algebras in a coordinate free form. The construction is done in an orthogonal basis, and the algebras characterized by universality. The basic properties with short proofs provides a clear foundation for further development of the algebras.