Quaternionic structure

Template:Short description In mathematics, a quaternionic structure or Template:Math-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field.

A quaternionic structure is a triple Template:Math where Template:Math is an elementary abelian group of exponent Template:Math with a distinguished element Template:Math, Template:Math is a pointed set with distinguished element Template:Math, and Template:Math is a symmetric surjection Template:Math satisfying axioms

$\begin{array}{cc}\text{1.}& q(a,(-1)a)=1,\\ \text{2.}& q(a,b)=q(a,c)\iff q(a,bc)=1,\\ \text{3.}& q(a,b)=q(c,d)\Rightarrow \mathrm{\exists }x\mid q(a,b)=q(a,x),q(c,d)=q(c,x)\end{array}.$

Every field Template:Math gives rise to a Template:Math-structure by taking Template:Math to be Template:Math, Template:Math the set of Brauer classes of quaternion algebras in the Brauer group of Template:Math with the split quaternion algebra as distinguished element and Template:Math the quaternion algebra Template:Math.

References

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