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Nilpotent-by-abelian Lie algebras of type FPm

Published online by Cambridge University Press:  10 February 2010

J. R. J. GROVES  and
DESSISLAVA H. KOCHLOUKOVA
J. R. J. GROVES
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Parkville, Vic. 3010, Australia. e-mail: jrjg@unimelb.edu.au
DESSISLAVA H. KOCHLOUKOVA
Affiliation:
Department of Mathematics, University of Campinas (UNICAMP), Cx. P. 6065, 13083-970 Campinas, SP, Brazil. e-mail: desi@ime.unicamp.br
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Abstract

Let L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L/N. We discuss the relationship between the homological finiteness properties of V and those of L. In particular, we show that if L has type FPm and N has class c then V is 1 + c(m − 1)-tame (equivalently, the (1 + c(m − 1))th tensor power of V is finitely generated under the diagonal action of L/N).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 3 , May 2010 , pp. 429 - 437
Copyright
Copyright © Cambridge Philosophical Society 2010

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