论文标题
$ \ Mathfrak g \ _2 $的换向关系和Fano飞机的发病率几何形状
Commutation relations of $\mathfrak g\_2$ and the incidence geometry of the Fano plane
论文作者
论文摘要
我们继续对Fano Plane $ {\ cal f} $上的结构进行研究和分类，其双$ {\ cal f}^\ ast $涉及八元建设和lie代数$ \ mathfrak g_2（\ mathbb f）$ \ mathbb f $。这些是“组成因子”：$ {\ cal f} \ times {\ cal f} \ to \ { - 1，1，1，1 \} $，诱导八次乘法，和一个函数$Δ^\ ast：aut（aut（aut（{\ cal f}） \ in aut（{\ cal f}）$可以将八元的自动形态提升为iff $δ^\ ast（g，\ cdot）$是$ {\ cal f} $上的函数的radon变换。我们将$ aut（{\ cal f}）$在$ {\ cal f} $上的动作提升到非平凡的八倍覆盖$ aut（{\ cal f}）$上的动作，在$ \ hat $ \ hat {\ cal f} $ of $ {\ cal f} $上包含的双重覆盖率{\ cal f} $。这是对自动形态学的八氧化行动的延伸。最后，我们将$ \ Mathfrak g_2（\ Mathbb F）$的生成点配对与发电括号相关联，以$ {\ cal f} $和$ε$的发病率几何形状而言。
We continue our study and classification of structures on the Fano plane ${\cal F}$ and its dual ${\cal F}^\ast$ involved in the construction of octonions and the Lie algebra $\mathfrak g_2 (\mathbb F)$ over a field $\mathbb F$. These are a "composition factor": ${\cal F}\times {\cal F} \to\{-1, 1\}$, inducing an octonion multiplication, and a function $δ^\ast : Aut({\cal F}) \times {\cal F}^\ast \to \{-1, 1\}$ such that $g \in Aut({\cal F})$ can be lifted to an automorphism of the octonions iff $δ^\ast(g, \cdot)$ is the Radon transform of a function on ${\cal F}$. We lift the action of $Aut({\cal F})$ on ${\cal F}$ to the action of a non-trivial eight-fold covering $Aut({\cal F})$ on a twofold covering $\hat {\cal F}$ of ${\cal F}$ contained in the octonions. This extends tautologically to an action on the octonions by automorphism. Finally, we associate to incident point-line pairs a generating set of $\mathfrak g_2 (\mathbb F)$ and express brackets in terms of the incidence geometry of ${\cal F}$ and $ε$.
