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首页> 外文期刊>Jahresbericht der Deutschen Mathematiker-Vereinigung >Francis Buekenhout and Arjeh M. Cohen: 'Diagram Geometry'
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Francis Buekenhout and Arjeh M. Cohen: 'Diagram Geometry'

机译：弗朗西斯·布肯豪特（Francis Buekenhout）和阿赫·科恩（Arjeh M. Cohen）：“图几何”

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摘要

Diagram geometry has its roots in the work of Tits during the 1950s when he was searching for suitable axioms for the combinatorial structures which are nowadays known as buildings. The starting point was the observation that the Dynkin-diagram A_n has a natural interpretation in terms of the n -dimensional projective geometry. The Dynkin-diagram A_n is associated with the semi-simple group SL_(n+1) which has a natural action on n-dimensional projective space. Tits' programme was to look for similar geometries for the semi-simple groups of the remaining types. In view of the interpretation of the diagram A_n for n-dimensional projective geometry, it was natural to guess that the diagram should be a central ingredient in the definition of the desired objects. This led Tits to the definition of a geometry over a Dynkin-diagram. Later on, however, he realized that the Bruhat decomposition provides a more direct approach to the link between Lie-theory and combinatorics. This had the effect that diagram geometries only played a minor role in Tits' theory of buildings and BN-pairs. Although one can consider buildings as diagram geometries, there is a priori no diagram required in their definition. Moreover, the slightly more abstract notion of a building turned out to be much more efficient. As a consequence the theory of diagram geometries was not pushed further for a while.
机译：图解几何学起源于1950年代Tits的工作，当时他正在为组合结构寻找合适的公理，如今这种组合被称为建筑。起点是观察到Dynkin图A_n就n维射影几何具有自然解释。 Dynkin图A_n与对n维投影空间具有自然作用的半简单组SL_（n + 1）关联。 Tits的程序是为其余类型的半简单组寻找相似的几何形状。鉴于对n维投影几何图形A_n的解释，很自然地猜出该图形应该是所需对象定义中的核心要素。这导致Tit提出了Dynkin-diagram上几何的定义。然而，后来他意识到布鲁汉特分解为李理论和组合论之间的联系提供了更直接的方法。结果是，图的几何形状在Tits的建筑物和BN对理论中仅扮演了次要角色。尽管可以将建筑物视为图的几何形状，但是在定义时不需要先验图。而且，建筑物的抽象概念稍微更为有效。结果，图几何学的理论并没有被进一步推动。

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