论文标题
约旦网和约旦网的几何形状
The Geometries of Jordan nets and Jordan webs
论文作者
论文摘要
Jordan Net（分别Web）是尺寸的Unital Jordan代数的嵌入，$ 3 $（分别为$ 4 $）$ \ MATHBB {s}^n $的对称$ n \ times n $矩阵。我们研究Jordan Nets和Web的几何形状：我们将Jordan Nets（webs）的一致性分类为$ \ Mathbb {s}^n $，for $ n \ leq 7 $（resp。$ n \ n \ leq 5 $），我们发现这些Orbits和这些Orbits和列出了这些障碍物之间的变性。对于$ \ Mathbb {s}^n $的Jordan Nets而言，对于$ n \ leq5 $，这些障碍物表明我们的退化列表已完成。对于$ n = 6 $，仍然不确定一个变性。 为了进一步探索，我们使用了一种算法，该算法在数值上是否存在两个轨道之间的变性。我们使用所有已知的退化和障碍物验证了该算法，然后用它来计算$ \ Mathbb {s}^7 $和Jordan Web的Jordan Nets之间的变性，而Jordan Web则在$ \ Mathbb {s}^n $中$ n = 4,5 $。
A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension $3$ (resp. $4$) into the space $\mathbb{S}^n$ of symmetric $n\times n$ matrices. We study the geometries of Jordan nets and webs: we classify the congruence-orbits of Jordan nets (resp. webs) in $\mathbb{S}^n$ for $n\leq 7$ (resp. $n\leq 5$), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in $\mathbb{S}^n$ for $n\leq5$, these obstructions show that our list of degenerations is complete. For $n=6$, the existence of one degeneration is still undetermined. To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions, and then used it to compute the degenerations between Jordan nets in $\mathbb{S}^7$ and Jordan webs in $\mathbb{S}^n$ for $n=4,5$.
