Metacommutation in Central Simple Algebras

TL;DR

This paper explores how non-commutative multiplication in central simple algebras causes permutations of primes in factorizations, generalizing known quaternion order results to broader algebraic contexts.

Contribution

It extends the understanding of prime permutation structures from quaternion orders to general orders in central simple algebras over global fields.

Findings

Permutation structures depend on non-commutativity

Cycle structures and fixed points are characterized

Generalization from quaternion to central simple algebras

Abstract

In a quaternion order of class number one, an element can be factored in multiple ways depending on the order of the factorization of its reduced norm. The fact that multiplication is not commutative causes an element to induce a permutation on the set of primes of a given reduced norm. We discuss this permutation and previously known results about the cycle structure, sign, and number of fixed points for quaternion orders. We generalize these results to other orders in central simple algebras over global fields.