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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Free ideals of one-relator graded Lie algebras


Author: John P. Labute
Journal: Trans. Amer. Math. Soc. 347 (1995), 175-188
MSC: Primary 17B01; Secondary 17B70
DOI: https://doi.org/10.1090/S0002-9947-1995-1282891-0
MathSciNet review: 1282891
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Abstract: In this paper we show that a one-relator graded Lie algebra $\mathfrak {g} = L/(r)$, over a principal ideal domain $K$, has a homogeneous ideal $\mathfrak {h}$ with $\mathfrak {g}/\mathfrak {h}$ a free $K$-module of finite rank if the relator $r$ is not a proper multiple of another element in the free Lie algebra $L$. As an application, we deduce that the center of a one-relator Lie algebra over $K$ is trivial if the rank of $L$ is greater than two. As another application, we find a new class of one-relator pro-$p$-groups which are of cohomological dimension $2$.


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Article copyright: © Copyright 1995 American Mathematical Society