Non-commutative frieze patterns over quaternion algebras and other normed division rings

Abstract

Non-commutative friezes have been introduced by Berenstein and Retakh and studied further by the authors. In this paper we consider non-commutative friezes over normed division rings, like for instance Hamilton's quaternions or more general quaternion algebras. We address the fundamental question in the theory of friezes of whether over a certain subset there are finitely or infinitely many non-commutative friezes (with 1's on the boundary) for any height. As an application of a theorem bounding the norm of quiddity entries we deduce that for every norm-finite subset of a normed division ring there are only finitely many such non-commutative friezes for every height. In particular this result applies to the Lipschitz quaternions and the Hurwitz quaternions of Hamilton's quaternions. We then study more generally non-commutative friezes over Lipschitz subrings of non-split quaternion algebras $(a,b)_{\mathbb{Q}}$. We determine the frieze subrings for all $a,b<0$, and as a consequence we see that all such non-commutative friezes are known if $a\le -4$ and $b\le -4$.