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 $G \times G \to Q$ satisfying axioms

{\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 $(a, b) F$ .

References

Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.