关于幂零与下中心同态映射的Quandle同态
On Nilpotent and Hypocentral Quandle Homomorphisms
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中文总结 AI 辅助
本文为quandle满射同态定义幂零与下中心概念，证明其等价于覆盖同态复合及右正交性，建立正交分解系统，并给出约化同态的幂零充分条件。
中文摘要 AI 辅助
我们引入了满射quandle同态的幂零性概念，该概念相对化了Darné最近定义的quandle的幂零性。一个满射同态被称为幂零的，如果其相对内自同构群相对于源的内自同构群是幂零的。我们证明了这个条件同样可以通过相对换位群和相对伴随群来描述，并且它等价于有限个覆盖同态的复合。将相对下中心列扩展到超限序数，我们随后引入了下中心同态，并将它们刻画为与强连通同态右正交的满射。作为推论，强连通同态和下中心同态在quandle范畴中形成了满射的正交分解系统。最后，我们还引入了约化同态和算子约化同态，它们相对化了约化quandle，并给出了这样的同态是幂零的充分条件。
英文摘要
We introduce a notion of nilpotency for surjective quandle homomorphisms, which relativizes the nilpotency of quandles recently defined by Darné. A surjective homomorphism is called nilpotent if its relative inner automorphism group is nilpotent relative to the inner automorphism group of the source. We show that this condition can equally be described through the relative transvection group and the relative adjoint group, and that it is equivalent to being a finite composite of covering homomorphisms. Extending the relative lower central series to transfinite ordinals, we then introduce hypocentral homomorphisms and characterize them as the surjections that are right orthogonal to the strongly connected homomorphisms. As a consequence, the strongly connected and the hypocentral homomorphisms form an orthogonal factorization system for the surjections in the category of quandles. Finally, we also introduce reduced and operator-reduced homomorphisms, which relativize reduced quandles, and give sufficient conditions for such a homomorphism to be nilpotent.
发表机构
- Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University（大阪中央高等数学研究所，大阪公立大学）
- Department of Mathematics, School of Science and Engineering, Waseda University（早稻田大学理工学术院数学系）
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