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 1. q ( a , ( − 1 ) a ) = 1 , 2. q ( a , b ) = q ( a , c ) ⇔ q ( a , b c ) = 1 , 3. q ( a , b ) = q ( c , d ) ⇒ ∃ x ∣ q ( a , b ) = q ( a , x ) , q ( c , d ) = q ( c , x ) . {\displaystyle {\begin{aligned}{\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{aligned}}.} 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.