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Adjoint action of a finite loop space. II

Published online by Cambridge University Press:  14 November 2011

Norio Iwase  and
Akira Kono
Norio Iwase
Affiliation:
Graduate School of Mathematics, Kyushu University Ropponmatsu, Fukuoka 810, Japan
Akira Kono
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan
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Extract

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 > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 773 - 785
Copyright
Copyright © Royal Society of Edinburgh 1999

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