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On the component group of the automorphism group of a Lie group

Part of: Locally compact groups and their algebras Lie groups

Published online by Cambridge University Press:  09 April 2009

P. B. Chen  and
T. S. Wu
P. B. Chen
Affiliation:
Department of Mathematics and Computer Science, John Carroll University, University Heights, OH 44118, USA
T. S. Wu
Affiliation:
Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA
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Abstract

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Let G be a Lie group, Go the connected component of G that contains the identity, and Aut G the group of all topological automorphisms of G. In the case when G/Go is finite and G has a faithful representation, we obtain a necessary and sufficient condition for G so that Aut G has finitely many components in terms of the maximal central torus in Go.

MSC classification

Secondary: 22E15: General properties and structure of real Lie groups 22D45: Automorphism groups of locally compact groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 3 , December 1994 , pp. 365 - 385
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Chevalley, C., Theory of Lie groups, Vol. I (Princeton Univ. Press, Princeton, 1946).Google Scholar
[2]Chevalley, C., Theorie des groupes de Lie, Vol. III (Hermann, Paris, 1955).Google Scholar
[3]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley, New York, 1962).Google Scholar
[4]Dani, S. G., ‘On automorphism groups of connected Lie groups’, Manuscripta Math. 74 (1992), 445–452.CrossRefGoogle Scholar
[5]Goto, M. and Wang, H. C., ‘Non-discrete uniform subgroups of semisimple Lie groups’, Math. Ann. 198 (1972), 259–286.CrossRefGoogle Scholar
[6]Hochschild, G., ‘The automorphism group of a Lie group’, Trans. Amer. Math. Soc. 72 (1952), 209–216.Google Scholar
[7]Hochschild, G., The structure of Lie groups (Holden-Day, San Francisco, 1965).Google Scholar
[8]Hochschild, G., ‘On representing analytic groups with their automorphisms’, Pacific J. Math. 78 (1978), 333–336.CrossRefGoogle Scholar
[9]Hungerford, T. W., Algebra (Holt, New York, 1974).Google Scholar
[10]Isaacs, I. M., Character theory of finite groups (Academic Press, New York, 1976).Google Scholar
[11]Iwasawa, K., ‘On some types of topological groups’, Ann. of Math. (2) 54 (1949), 507–558.CrossRefGoogle Scholar
[12]Lee, D. H., ‘Supplements for the identity component in locally compact groups’, Math. Z. 104 (1968), 28–49.CrossRefGoogle Scholar
[13]Lee, D. H. and Wu, T. S., ‘On faithful representations of the holomorph of Lie groups’, Math. Ann. 275 (1986), 521–527.CrossRefGoogle Scholar
[14]Matsumoto, H., ‘Quelques remarques sur les groupes de Lie’, J. Math. Soc. Japan 16 (1964), 419–449.Google Scholar
[15]Ribenboim, P., Algebraic numbers (Wiley, New York, 1972).Google Scholar
[16]Wigner, D., ‘On the automorphism group of a Lie group’, Proc. Amer. Math. Soc. 45 (1974), 140–143.CrossRefGoogle Scholar
[17]Wigner, D., ‘Erratum to ‘On the automorphism group of a Lie group”, Proc. Amer. Math. Soc. 60 (1976), 376.Google Scholar