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Subalgebras of free restricted Lie algebras

Published online by Cambridge University Press:  17 April 2009

R.M. Bryant ,
L.G. Kovács  and
Ralph Stöhr
R.M. Bryant
Affiliation:
School of Mathematics, University of Manchester, PO Box 88, Manchester M60 1QD, United Kingdom, e-mail: roger.bryant@manchester.ac.uk, ralph.stohr@manchester.ac.uk
L.G. Kovács
Affiliation:
School of Mathematics, University of Manchester, PO Box 88, Manchester M60 1QD, United Kingdom, e-mail: roger.bryant@manchester.ac.uk, ralph.stohr@manchester.ac.uk
Ralph Stöhr
Affiliation:
Australian National University, Canberra ACT 0200, Australia, e-mail: kovacs@maths.anu.edu.au
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A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 1 , August 2005 , pp. 147 - 156
Copyright
Copyright © Australian Mathematical Society 2005

References

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