关于同构淬表的嵌入
On embeddings of homogeneous quandles
AI总结：
本文研究了同构淬表的嵌入问题，提出了一个必要且充分条件，并应用该结果重新诠释了核心淬表的嵌入，同时构建了多个几何示例的显式嵌入。
摘要：
在本文中，我们研究了同构淬表的嵌入问题。我们给出了一个必要且充分条件，即从由淬表三元组（G,H,σ）关联的同构淬表到群的共轭淬表的淬表同态是嵌入的条件。这一结果是对Dhanwani、Raundal和Singh为广义Alexander淬表提出的嵌入定理的推广。作为主定理的应用，我们重新诠释了Bergman对核心淬表的嵌入，以同构淬表的框架，并构建了多个几何示例的显式嵌入，包括无向和有向的格拉斯曼淬表以及来自对称空间的球面旋转淬表。
英文摘要：
In this paper, we study the embedding problem of homogeneous quandles. We give a necessary and sufficient condition under which a quandle homomorphism from the homogeneous quandle associated with a quandle triplet $(G,H,σ)$ into a conjugation quandle of a group is an embedding. This result provides a generalization of the embedding theorem of Dhanwani, Raundal and Singh for generalized Alexander quandles. As applications of the main theorem, we reinterpret Bergman's embedding of core quandles in the framework of homogeneous quandles, and construct explicit embeddings of several geometric examples, including unoriented and oriented Grassmann quandles and rotation quandles of $S^2$ arising from symmetric spaces.
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