Quaternionic structure

In mathematics, a quaternionic structure or $Q$-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field.

A quaternionic structure is a triple $(G, Q, q)$ where $G$ is an elementary abelian group of exponent $2$ with a distinguished element $−1$, $Q$ is a pointed set with distinguished element $1$, and $q$ is a symmetric surjection $G×G → Q$ satisfying axioms


 * $$\begin{align}\text{1.} \quad &q(a,(-1)a) = 1,\\

\text{2.} \quad &q(a,b) = q(a,c) \Leftrightarrow q(a,bc) = 1,\\ \text{3.} \quad &q(a,b) = q(c,d) \Rightarrow \exists x\mid q(a,b) = q(a,x), q(c,d) = q(c,x)\end{align}.$$

Every field $F$ gives rise to a $Q$-structure by taking $G$ to be $F^{∗}/F^{∗2}$, $Q$ the set of Brauer classes of quaternion algebras in the Brauer group of $F$ with the split quaternion algebra as distinguished element and $q(a,b)$ the quaternion algebra $(a,b)_{F}$.