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

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

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.