Subalgebras of free restricted Lie algebras

R. M. Bryant*, L. G. Kovács, Ralph Stöhr

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    8 Citations (Scopus)


    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. Copyright Clearance Centre, Inc.

    Original languageEnglish
    Pages (from-to)147-156
    Number of pages10
    JournalBulletin of the Australian Mathematical Society
    Issue number1
    Publication statusPublished - Aug 2005


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