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

$\begin{array}{cc}\text{1.}& q(a,(-1)a)=1,\\ \text{2.}& q(a,b)=q(a,c)\iff q(a,bc)=1,\\ \text{3.}& q(a,b)=q(c,d)\Rightarrow \mathrm{\exists }x\mid q(a,b)=q(a,x),q(c,d)=q(c,x)\end{array}.$

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.