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Transactions of the American Mathematical Society

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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
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