Introduction

The free algebra on a set, and the tensor and exterior algebras of a vector space are constructed. Emphasis is placed on the exterior algebra of a finite dimensional vector space and its applications to determinants. Afterwards an alternate independent development of the exterior algebra is given as the quotient of a tensor algebra.

Why is the exterior and to a lesser extent the tensor algebra given such prominence in this chapter as opposed to other algebras? The exterior algebra appears in differential geometry not only as the algebra of differential forms but also from skew symmetric tensors. There seems to be an abundant supply of the latter perhaps because the Lie product [x, y] = xy – yx is skew symmetric in x and y, and various Lie operations (e.g. the Lie derivative) and Lie algebras appear unavoidably in differential geometry and in physics as well. Some constructions and concepts in physics can be formulated more precisely, more easily, and be better understood if one has the tensor and exterior algebras at one's disposal. There is a lot of interesting and useful material here that is not covered in this chapter, which hopefully will serve as an introduction and invitation to further study.

This chapter takes place in the category of algebras with identity over a fixed ground field F and identity preserving homomorphisms. The identity element of a subalgebra is required to be the identity of the big algebra containing it.