Embeddings of Quadratic Spaces

Abstract

We introduce the concept of an embedding of a quadratic space in an associative algebra. Using Clifford Algebras we derive some fundamental properties that any embedding should satisfy. Conversely, there is a simple description of the Clifford Algebra and the corresponding Spin groups in terms of the algebra in which the quadratic space is embedded. Though Clifford Algebras have been studied in detail, they may not always be easy to work with. Sometimes it may be useful to switch to a more concrete embedding to study low dimensional Spin and Epin (or Elementary Spin) groups.