关于Kac-Moody群的拓扑
On the Topology of Kac-Moody groups
摘要：
研究了与Kac-Moody群相关的空间的拓扑。给定一个复数上的可裂Kac-Moody群，设K表示具有极大环面T的酉形式，其正规化子为N（T）.本文研究旗流形K/T作为Nil-Hecke环上的模的上同调，以及K作为霍普夫代数的（上）同调.特别地，如果F是一个正特征的域，我们证明了H_*（K，F）是一个双生成代数，且H^*（K，F）是双生成代数的唯一条件是K是一个紧李群.我们还研究了分类空间BK的稳定同伦型，证明了它是分类空间BN（T）的收缩核。我们用秩为2的Kac-Moody群的例子来说明我们的结果。
英文摘要：
We study the topology of spaces related to Kac-Moody groups. Given a split Kac-Moody group over the complex numbers, let K denote the unitary form with maximal torus T having normalizer N(T). In this article we study the cohomology of the flag manifold K/T, as a module over the Nil-Hecke ring, as well as the (co)homology of K as a Hopf algebra. In particular, if F is a field of positive characteristic, we show that H_*(K,F) is a finitely generated algebra, and that H^*(K,F) is finitely generated only if K is a compact Lie group . We also study the stable homotopy type of the classifying space BK and show that it is a retract of the classifying space BN(T). We illustrate our results with the example of rank two Kac-Moody groups.
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