奥库博代数的整序与$E_8$格的幂等几何
Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice
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中文总结 AI 辅助
该研究以$E_8$序为共同整支撑分析三种代数结构的幂等元差异，揭示其对$E_8$根系的极化组织方式，为$E_8$相关分解提供算术解释。
中文摘要 AI 辅助
我们将Coxeter-Dickson $E_8$序作为八元数、准八元数及紧致实奥库博乘积的共同整支撑。这三种代数结构具有相同的加法格、相同的正合成范数，因此拥有相同的240个范数为1的向量，即$E_8$的根与Gosset多面体$4_{21}$的顶点。不过它们的乘法结构不同，这种差异在幂等元中已显现：在相同的240个根中，分别存在1个、57个和12个非零整幂等元。我们证明这三个计数以三种日益极化的方式组织共同根系：八元数乘积未分离完整的$E_8$几何；57个准八元数幂等元再现$E_7$接触分解$1+56+126+56+1$；12个奥库博幂等元分裂为四个相互正交的定向$A_2$三角形，从而正则确定$A_2^4$子系统。选取一个三角形作为外部$A_2$，得到$A_2+E_6$魔术星，剩余9个幂等元确定三重态子系统$A_2^3/\u2282 E_6$。这直接从整幂等元给出$E_8/\u2283 E_7$与$E_8/\u2283 E_6+A_2$分解的算术解释。
英文摘要
We study the Coxeter-Dickson $E_8$ order as a common integral support for the octonionic, para-octonionic and compact real Okubo products. The three algebra structures have the same additive lattice, the same positive composition norm and hence the same $240$ norm-one vectors, namely the roots of $E_8$ and the vertices of the Gosset polytope $4_{21}$. Their multiplicative structures are nevertheless different and this difference is already visible in the idempotents: among the same $240$ roots one finds respectively $1$, $57$ and $12$ nonzero integral idempotents. We show that these three counts organize the common root system in three increasingly polarized ways. The octonionic product leaves the full $E_8$ geometry unseparated; the $57$ para-octonionic idempotents reproduce the $E_7$ contact decomposition $1+56+126+56+1$; the $12$ Okubo idempotents split into four mutually orthogonal oriented $A_2$ triangles and hence determine canonically an $A_2^4$ subsystem. Choosing one triangle as the external $A_2$ then yields the $A_2+E_6$ Magic Star, while the remaining nine idempotents determine the trinification subsystem $A_2^3\subset E_6$. This provides an arithmetic interpretation of the $E_8\supset E_7$ and $E_8\supset E_6+A_2$ decompositions directly from integral idempotents.
