括号一致性的几何图形
Geometry from Brackets Consistency
摘要：
我们认为括号的一致性是一个“支柱的几何”，即一个基础的方法，其他支柱是1）欧几里德的建设性的方法，2）代数方法，3）投影的方法，和4）几何自同构群“埃尔兰根”的方法。我们继续通过一个独特的支柱5）- Killing的计算路线埃尔兰根的群-首先挑选出随后的几何重要括号子代数作为一致的可能性。其次，通过考虑由不同的广义Killing方程产生的生成元，将括号一致地组合起来，恢复仿射共形和射影共形的替代方案。第三，我们不再完全依赖于Killing's Pillar，而是用单纯的多项式ansätaze作为生成元的来源。即使在这种情况下-要求（谎言）括号一致性-足以恢复几何的概念，如下面的替代只是退出随后的括号代数所示，因此被宣布为支柱6。本文的进一步动机，首先，在理论物理学的重新制定上运行经典约束代数结构一致性的狄拉克算法的逐步积累的成功。其次，这是由括号一致性本身足以超越任何上下文拥有这样的括号。第三，我们导出了上自同构群$d \geq 3$平坦几何中的共形或射影的选择。在狄拉克算法的应用中，这两个顶级几何以一种现在已经熟悉的方式，a）而且作为一个仅仅是代数二次方程的根而并排出现。b）这个方程产生于要求生成元的一般代数的自括号强为零。c）因为，如果不是这样的话，一个无限的发生器级联将随之而来，这样的假设将不支持一个有限的形状理论。
英文摘要：
We argue for Brackets Consistency to be a `Pillar of Geometry', i.e. a foundational approach, other Pillars being 1) Euclid's constructive approach, 2) the algebraic approach, 3) the projective approach, and 4) the geometrical automorphism groups `Erlangen' approach. We proceed via a distinct Pillar 5) - Killing's computational route to Erlangen's groups - firstly by picking out the ensuing geometrically significant brackets subalgebras as consistent possibilities. Secondly, by considering which generators, arising piecemeal from different generalized Killing equations, combine brackets-consistently, recovering the affine-conformal and projective-conformal alternatives. Thirdly, we cease to rest on Killing's Pillar altogether, replacing it by mere polynomial ansätaze as source of generators. Even in this case - demanding (Lie) Brackets Consistency - suffices to recover notions of geometry, as indicated by the below alternative just dropping out of the ensuing brackets algebra, and is thus declared to be Pillar 6. The current article is further motivated, firstly, by the gradually accumulating successes in reformulating Theoretical Physics to run on the Dirac Algorithm for classical constraint algebraic structure consistency. Secondly, this is sufficiently driven by Brackets Consistency alone to transcend to whichever context possesses such brackets. Thirdly, we derive the conformal-or-projective alternative in top-automorphism-group $d \geq 3$ Flat Geometry. In a manner by now familiar in Dirac Algorithm applications, a) these two top geometries moreover arise side by side as roots of a merely algebraic quadratic equation. b) This equation arises from requiring that the self-bracket of a general ansatz for generators be strongly zero. c) For, if not, an infinite cascade of generators would follow, by which the ansatz would not support a finite Shape Theory.
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