A theorem of tits, normalizers of maximal tori and fibrewise Bousfield-Kan completions
A theorem of tits, normalizers of maximal tori and fibrewise Bousfield-Kan completions
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山雀定理、最大环面标准化和纤维布斯菲尔德-坎完成
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发表时间：
1999
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通讯作者：
F. Neumann
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作者：
F. Neumann
利用Tits关于连通紧半单李群的极大环面的正规化子的生成元和关系表示的一个定理，给出了伴随的正规化子群扩张分裂的几个等价条件，并用p-进纤维同伦理论解释了它们.§0.前言紧连通半单李群G的同构类型完全由Curtis，Wiederhold和Williams[C-W-W]所证明的极大环面正规化子的同构类型决定。这一点后来由Notbohm推广到任何紧连通李群[N]。在他们的经典论文中，Curtis，Wiederhold和Williams也研究了当群扩张0--T(G)--N(G)-W(G)-i是分裂扩张且正规化子N(G)完全由Weyl群W(G)在极大环面Tg上的作用所决定时的相关问题。利用Tits[T2]的一个定理，给出了正规化子的生成元和关系的显式刻画，他们可以逐个地判定上述正规化子序列对于哪个单李群是分裂精确的。在这篇注记中，我们利用Tits定理来解释上述正规化子群扩张的分裂，它是通过分类空间的伴随纤维的p-进Bousfield-Kan完备化[B-K]来实现的。它由K.Saito于1999年11月9日传达。1991年数学科目分类。首页--期刊主要分类--期刊细介绍--期刊题录与文摘--期刊详细文摘内容3-5，D-37073，德国哥廷根，电子邮件：Neumann®cfgauss.uni-math.gwdg.de
We use a theorem of Tits on the presentation of the normalizer of a maximal torus of a connected compact semisimple Lie group in terms of generators and relations to give several equivalent conditions for the splitting of the associated normalizer group extension and interprete them in terms of p-adic fibrewise homotopy theory. §0. Introduction The isomorphism type of a compact connected semisimple Lie group G is completely determined by the isomorphism type of the normalizer of the maximal torus JG) as it was shown by Curtis, Wiederhold and Williams [C-W-W]. This was generalized much later by Notbohm for any compact connected Lie group [N]. In their classical paper Curtis, Wiederhold and Williams also studied the related question when the group extension 0 -» T(G) -»N(G) ^ W(G) -» I is a split extension and the normalizer N(G) is completely determined by the action of the Weyl group W(G) on the maximal torus TG Using a theorem of Tits [T2] giving an explicit description of the normalizer in terms of generators and relations, they could decide case-by-case for which simple Lie groups the above normalizer sequence is split exact. In this note we use the theorem of Tits to interprete the splitting of the above normalizer group extension in terms of fibrewise p-adic Bousfield-Kan completion [B-K] of the associated fibration of classifying spaces. It turns Communicated by K. Saito, November 9, 1999. 1991 Mathematics Subject Classification. 20 F 55, 20 G 20, 20 J 06, 22 E 15, 57 T 10 Mathematisches Institut, University of Gottingen, Bunsenstr. 3-5, D-37073 Gottingen, Germany e-mail: neumann® cfgauss.uni-math.gwdg.de