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Centre-by-metabelian Lie algebras

Published online by Cambridge University Press:  09 April 2009

M. R. Vaughan-Lee
M. R. Vaughan-Lee
Affiliation:
University of Queensland, Australia
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If V is a variety of metabelian Lie algebras then V has a finite basis for its laws [3]. The proof of this result is similar to Cohen's proof that varieties of metabelian groups have the finite basis property [1]. However there are centre-by-metabelian Lie algebras of characteristic 2 which do not have a finite basis for their laws [4] this contrasts with McKay's recent result that varieties of centre-by-metabelian groups do have the finite basis property [2]. The rollowing theorem shows that once again “2” is the odd man out.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 3 , May 1973 , pp. 259 - 264
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Cohen, D. E., ‘On the laws of a metabelian variety’, J. Algebra, 5 (1967), 267–273.CrossRefGoogle Scholar
[2]Susan, McKay, Some problems in group theory (Oxford D. Phil. thesis, 1970).Google Scholar
[3]Vaughan-Lee, M. R., Some varieties of Lie algebras (Oxford D. Phil. thesis, 1968).Google Scholar
[4]Vaughan-Lee, M. R., ‘Varieties of Lie algebras’, Quart. J. Math. Oxford Ser. 21 (1970), 297–308.CrossRefGoogle Scholar
[5]Vaughan-Lee, M. R., ‘Abelian-by-nilpotent varieties’, Quart. J. Math. Oxford Ser. 21 (1970), 193–202.CrossRefGoogle Scholar