TY - JOUR

T1 - Adjoint action of a finite loop space. II

AU - Iwase, Norio

AU - Kono, Akira

N1 - Funding Information:
The authors thank Katsuhiko Kuribayashi for valuable conversations and emails without which this work could not be completed and University of Aberdeen for its hospitality during the first author's stay in Aberdeen. This research was partly supported by Grant-in-Aid for Scientific Research (C)08640125 from The Ministry of Science, Sports and Culture.

PY - 1999

Y1 - 1999

N2 - Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo-Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j : G → BΛG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

AB - Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo-Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j : G → BΛG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

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U2 - 10.1017/S0308210500013135

DO - 10.1017/S0308210500013135

M3 - Article

AN - SCOPUS:22644449242

VL - 129

SP - 773

EP - 785

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 4

ER -