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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Serial subalgebras of finitary Lie algebras


Authors: Felix Leinen and Orazio Puglisi
Journal: Proc. Amer. Math. Soc. 129 (2001), 45-51
MSC (1991): Primary 17B65, 17B50
Published electronically: July 27, 2000
MathSciNet review: 1694350
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Abstract:

A Lie subalgebra $L$ of ${\mathfrak{gl}_{{\mathbb{K}}}(V)}$ is said to be finitary if it consists of elements of finite rank. We show that, if $L$ acts irreducibly on $V$, and if $V$ is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of $L$ acts irreducibly on $V$ too. When $\operatorname{Char} \mathbb{K}\neq 2$, it follows that the locally solvable radical of such $L$ is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic $\neq 2$ are hyperabelian.


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Additional Information

Felix Leinen
Affiliation: Fachbereich 17 – Mathematik, Johannes Gutenberg–Universität Mainz, D–55099 Mainz, Germany
Address at time of publication: Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU, United Kingdom
Email: Leinen@mathematik.uni-mainz.de, F.A.Leinen@ncl.ac.uk

Orazio Puglisi
Affiliation: Dipartimento di Matematica, Università degli Studi di Trento, I–38050 Povo (Trento), Italy
Email: puglisi@alpha.science.unitn.it

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05496-4
PII: S 0002-9939(00)05496-4
Keywords: Lie algebra, finitary endomorphism, serial subalgebra, locally solvable radical, Hirsch-Plotkin radical
Received by editor(s): September 3, 1998
Received by editor(s) in revised form: March 22, 1999
Published electronically: July 27, 2000
Communicated by: Roe Goodman
Article copyright: © Copyright 2000 American Mathematical Society