四元数代数及其他赋范可除环上的非交换花环图案
Non-commutative frieze patterns over quaternion algebras and other normed division rings
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中文总结 AI 辅助
本文研究赋范可除环（如四元数代数）上的非交换花环，证明范数有限子集上任意高度的花环有限，并确定非分裂四元数代数Lipschitz子环的花环子环。
中文摘要 AI 辅助
非交换花环由Berenstein和Retakh引入，并由作者进一步研究。本文考虑赋范可除环上的非交换花环，例如哈密顿四元数或更一般的四元数代数。我们处理花环理论中的一个基本问题：对于某一子集，在任意高度下，边界上具有1的非交换花环是有限多个还是无限多个。作为界定quiddity条目范数的一个定理的应用，我们推断：对于赋范可除环的每个范数有限子集，在任意高度下，这样的非交换花环仅有有限多个。特别地，该结果适用于哈密顿四元数中的Lipschitz四元数和Hurwitz四元数。然后，我们更一般地研究非分裂四元数代数$(a,b)_{\mathbb{Q}}$的Lipschitz子环上的非交换花环。我们确定了所有$a,b<0$的花环子环，并由此得出，若$a\le -4$且$b\le -4$，则所有这样的非交换花环都是已知的。
英文摘要
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 $b\le -4$.
发表机构
- Leibniz Universität Hannover（汉诺威莱布尼茨大学）
- Aarhus University（奥胡斯大学）
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