The Bruhat Decomposition
The Bruhat Decomposition
复制标题
DOI：
10.1007/978-1-4614-8024-2_27
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
D. Bump
中科院分区：
文献类型：
--
作者：
D. Bump
Bruhat分解在李群的历史中发现得相当晚，鉴于它的基本重要性，这是令人惊讶的。在此之前，埃雷斯曼发现了一个密切相关的细胞分解旗流形。Bruhat分解是由Tits在具有（B，N）对的群或Tits系统的概念中公理化的.这是Coxeter群概念的推广，实际上每个（B，N）都产生一个Coxeter群。我们在定理25.1之后已经指出，考克斯特群总是作用在单纯复形上，其几何形状与其性质密切相关。事实证明，具有（BN）对的群也作用于单纯复形，即山雀的建筑。我们将没有空间来讨论这个重要的概念，但请参阅Tits [163]和Abramenko和Brown [1]。
The Bruhat decomposition was discovered quite late in the history of Lie groups, which is surprising in view of its fundamental importance. It was preceded by Ehresmann’s discovery of a closely related cell decomposition for flag manifolds. The Bruhat decomposition was axiomatized by Tits in the notion of aGroup with(B,N)pairorTits’ system. This is a generalization of the notion of a Coxeter group, and indeed every (B,N) gives rise to a Coxeter group. We have remarked after Theorem 25.1 that Coxeter groups always act on simplicial complexes whose geometry is closely connected with their properties. As it turns out a group with (BN) pair also acts on a simplicial complex, theTits’ building. We will not have space to discuss this important concept but see Tits [163] and Abramenko and Brown [1].