Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-03-01T10:20:08.447Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected]
L.G. Kovács
Affiliation:
School of Mathematics, University of Manchester, PO Box 88, Manchester M60 1QD, United Kingdom, e-mail: [email protected], [email protected]
Ralph Stöhr
Affiliation:
Australian National University, Canberra ACT 0200, Australia, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Bakhturin, Yu. A., Identical relations in Lie algebras, (Russian) (Nauka, Moscow, 1985). English translation: (VNU Science Press, Utrecht, 1987).Google Scholar
[2]Bryant, R.M. and Stöhr, R., ‘On the module structure of free Lie algebras’, Trans. Amer. Math. Soc. 352 (2000), 901–934.CrossRefGoogle Scholar
[3]Bryant, R.M., Kovács, L.G. and Stöhr, R., ‘Lie powers of modules for groups of prime order’, Proc. London Math. Soc. (3) 84 (2002), 343–374.CrossRefGoogle Scholar
[4]Bourbaki, N., Lie groups and Lie algebras, (Part I: Chapters 1–3) (Hermann, Paris, 1975).Google Scholar
[5]Jacobson, N., Lie algebras (Interscience, New York, 1962).Google Scholar
[6]Kovács, L.G. and Stöhr, R., ‘Lie powers of the natural module for GL (2)’, J. Algebra 229 (2000), 435–462.CrossRefGoogle Scholar
[7]Kukin, G.P., ‘Subalgebras of free Lie p-algebras’, (Russian), Algebra i Logika 11 (1972), 535–550. English translation: Algebra Logic 11 (1972), 294–303.Google Scholar
[8]Reutenauer, C., Free Lie algebras (Clarendon Press, Oxford, 1993).CrossRefGoogle Scholar
[9]Shirshov, A.I., ‘Subalgebras of free Lie algebras’, (Russian), Mat. Sbornik N. S. 33 (1953), 441–452.Google Scholar
[10]Stöhr, R., ‘Restricted Lazard elimination and modular Lie powers’, J. Austral. Math. Soc. 71 (2001), 259–277.CrossRefGoogle Scholar
[11]Witt, E., ‘Die Unterringe der freien Lieschen Ringe’, Math. Z. 64 (1956), 195–216.CrossRefGoogle Scholar