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{align}\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| q(a,b) = q(a,x), q(c,d) = q(c,x)\end{align}.

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 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.

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