Clifford algebra, geometric algebra, and applications

TL;DR

This paper provides lecture notes on Clifford and geometric algebras, highlighting their theoretical foundations and diverse applications in mathematics and physics, including geometry, spinors, and graph theory.

Contribution

It offers a comprehensive introduction to Clifford algebra, emphasizing both its algebraic structure and wide-ranging applications across multiple disciplines.

Findings

Introduces Clifford algebra via tensor and geometric constructions

Explores applications in geometry, spinors, and graph theory

Provides algebraic tools for combinatorics and topology

Abstract

These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory.