On the K-theory of compact lie groups
On the K-theory of compact lie groups
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关于紧李群的 K 理论
DOI：
10.1016/0040-9383(65)90051-0
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发表时间：
1965
期刊：
影响因子：
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通讯作者：
M. Atiyah
中科院分区：
文献类型：
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作者：
M. Atiyah
我们还记得，对于任何(紧)空间X，K*(X)=K‘(X)@K’(X)，其中K‘(X)是X上的复向量丛的Grothendieck群，K’(X)是由X的悬挂上的丛或等价地由X到酉群U(N)的同伦映射类定义的。K*(X)是一个Z2-分次环，如果把表示p：G-+G‘(N)简单地看作一个连续映射，我们就得到了K’(G)的一个元素，记为b(P)。现在回想一下，G有1个基本的不可约表示p，pl，其最大权重I，，...I，形成一个角色分组的基础？霍奇金定理的极大环面T断言K*(G)是由j？(PI)，……(P[)生成的外代数。事实上，这个结果实际上有两个部分：(A)K*(G)没有挠率，
We recall that, for any (compact) space X, K*(X)= K’(X)@ K’(X) where K’(X) is the Grothendieck group of complex vector bundles over X and K’(X) is defined from bundles over the suspension of X or equivalently by homotopy classes of maps of X into the unitary group U (n)(for n large). K*(X) is a Z2-graded ring.If we consider a representation p: G-+ G’(n) simply as a continuous map we obtain an element of K’(G) which we shall denote by b (p). Now recall that G has 1 basic irreducible representations p,,.., pl, whose maximal weights I.,,.... i, form a basis for the character groupt? of the maximal torus T of G. Hodgkin’s theorem asserts that K*(G) is an exterior algebra generated by j?(pi),..../?(p [). In fact there are really two parts to this result:(A) K*(G) has no torsion,