Quaternionic structure

{{Short description|Axiomatic system in mathematics}}

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

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

:

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