Lesson plan
quaternionic structure
In mathematics, a quaternionic structure or -structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field. A quaternionic structure is a triple where is an elementary abelian group of exponent with a distinguished element , is a pointed set with distinguished element , and is a symmetric surjection 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\mid q(a,b) = q(a,x), q(c,d) = q(c,x)\end{align}. Every field gives rise to a -structure by taking to be , the set of Brauer classes of quaternion algebras in the Brauer group of with the split quaternion algebra as distinguished element and the quaternion algebra .
Article summary
In mathematics, a quaternionic structure or -structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field. A quaternionic structure is a triple where is an elementary abelian group of exponent with a distinguished element , is a pointed set with distinguished element , and is a symmetric surjection 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\mid q(a,b) = q(a,x), q(c,d) = q(c,x)\end{align}. Every field gives rise to a -structure by taking to be , the set of Brauer classes of quaternion algebras in the Brauer group of with the split quaternion algebra as distinguished element and the quaternion algebra .
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