Order in a quaternion algebra

Let $B$ be a quaternion algebra over a number field $K$ . An order in $B$ , in the sense of an $\mathcal O_K$ -order in an algebra, is a subring $\mathcal O\subseteq B$ containing $1$ and $\mathcal O_K$ , where $\mathcal O_K$ is the ring of integers, such that $\mathcal O$ is a finitely generated $\mathcal O_K$ -module and spans $B$ over $K$ .

Example

In the split algebra $M_2(K)$ , the subring $M_2(\mathcal O_K)$ is an order: the four matrix units give an integral module basis and span all matrices over $K$ .

Finiteness is not freeness

Over a general number ring, a full finitely generated module need not be free. Thus “has an integral basis of four elements over $\mathcal O_K$ ” would be an unnecessarily restrictive definition.

An order is maximal when it is contained in no larger order. Maximality is not required in the definition of an arithmetic Kleinian group.

Ideals and examples

The Lipschitz and Hurwitz orders give distinct $\mathbb Z$ -orders in the rational Hamilton algebra. Their reduced discriminants distinguish them.

One-sided fractional ideals must specify left or right multiplication. Their class set generally has no ideal-multiplication group law.

References

F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Transactions of the AMS 349 (1997). Author-hosted paper, §4, integral orders used in the arithmetic-group construction.