AI 中文总结
该研究依据欧几里得若尔当代数的秩，分类对称锥上完全正锥的面部暴露性，发现其分类结果与完全正锥的谱面性分类一致。
AI 中文摘要
我们依据相关欧几里得若尔当代数的秩，对对称锥上完全正锥的面部暴露性进行分类。当秩不超过2时，完全正锥是面部暴露的；当秩至少为5时，完全正锥不是面部暴露的。当秩为3或4时，面部暴露性并非完全由秩决定，我们针对每种情况给出了特征刻画。所得的面部暴露性分类实际上与完全正锥的谱面性对应分类一致。
英文摘要
We classify the facial exposedness of completely positive cones over symmetric cones in terms of the rank of the associated Euclidean Jordan algebras. The completely positive cones are facially exposed when the rank is at most $2$, but are not facially exposed when the rank is at least $5$. Facial exposedness is not completely determined by the rank when the rank is $3$ or $4$, but we provide a characterization in each case. The resulting classification of facial exposedness in fact agrees with the corresponding classification of spectrahedrality for completely positive cones.
