On the k-theory of Lie groups
On the k-theory of Lie groups
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关于李群的 k 理论
DOI：
10.1016/0040-9383(67)90010-9
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发表时间：
1967
期刊：
影响因子：
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通讯作者：
L. Hodgkin
中科院分区：
文献类型：
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作者：
L. Hodgkin
在Atiyah和Hirzebruch的文献[9]中，首先定义了(么正)K-理论，利用函子与经典群的关系给出了与李群相关的某些空间的特别简单的结果(与上同调相反)。对于李群本身来说，这个函子是否可以比通常的上同调理论更简单地描述是合理的，如果是这样的话，李群的K-理论的简单性是否可以揭示上同调中复杂的原因(参见[14]对这些问题的部分解释)。我们在这里只关心第一个问题(确实我们被迫从‘复杂的’上同调推导出‘简单的’K-理论)。当n，(G)=H，(G)为无挠时，我们得到了K*(G)(李群G的酉K-理论)的完整刻画。特别地，这涵盖了G是半单和单连通的情况。
IN THE paper [9] of Atiyah and Hirzebruch where (unitary) K-theory was first defined, the relation of the functor to classical groups was exploited to give particularly simple results (in contrast with cohomology) on certain spaces associated with Lie groups* lassifying spaces [9, $41 and some homogeneous spaces [9, $3.61. It is reasonable to ask whether for Lie groups themselves this functor can be described more simply than the usual cohomology theories, $ and if so whether the simplicity of the K-theory of Lie groups can throw light on the causes of complications in their cohomology (cf.[14] for some partial explanations of these).Our concern here is with the first of these problems only (indeed we are forced to deduce the ‘simple’K-theory from the ‘complicated’cohomology). The result we obtain is the following complete description of K*(G)(unitary K-theory of the Lie group G) when n,(G)= H,(G) is torsion-free. In particular this covers the case that G is semi-simple and simply-connected.