Embeddings of Quadratic Spaces

TL;DR

This paper explores embeddings of quadratic spaces into associative algebras, linking them to Clifford algebras, and provides simpler descriptions of involutions and Spin groups for easier study of low-dimensional cases.

Contribution

It introduces a new perspective on quadratic space embeddings, connecting them to Clifford algebras and simplifying the analysis of Spin groups.

Findings

Established properties of quadratic space embeddings

Linked embeddings to Clifford algebra structures

Provided simplified descriptions of Spin groups

Abstract

We introduce a concept of an embedding of a quadratic space in an associative algebra. The general properties of such embeddings are analyzed by linking it to the Clifford algebra. Conversely, there isa simple description of the standard involution and the 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.