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 from G×G → Q satisfying axioms

1. q(a,(−1)a) = 1;
2. q(a,b) = q(a,c) ⇔ q(a,bc) = 1;
3. q(a,b) = q(c,d) ⇒ ∃x . q(a,b) = q(a,x), q(c,d) = q(c,x).

Every field F gives rise to a Q-structure by taking G to be F^∗/F^∗2, Q the set 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.