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On the module structure of free Lie algebras. (English) Zbl 1017.17007

The authors study the free Lie algebra \(L\) over a field of non-zero characteristic \(p\) as a module for the cyclic group of order \(p\) acting on \(L\) by cyclically permuting the elements of a free generating set. For a given positive integer \(m\) and for \(n\geq 1\), let \(a(n)=(-\sum_{d:\;p|d|n}\mu(d)m^{n/d})/n\) where the sum ranges over all divisors \(d\) of \(n\) which are multiples of \(p\) and where \(\mu\) is the Möbius function. Further, for a group \(G\), a field \(K\), and a \(KG\)-module \(V\), let \(L(V)\) be the free Lie algebra on \(V\), \(L_n(V)\) the homogeneous component of \(L(V)\) of degree \(n\) and \(IG\) the augmentation ideal of \(KG\). The main result of the article is the following:
Theorem 1. Let \(p\) be a prime, \(G\) the cyclic group of order \(p\), \(K\) a field of characteristic \(p\) and \(V\) a free \(KG\)-module of finite dimension \(m\). Then \(L_n(V)\) is a direct sum of a free \(KG\)-module and \(a(n)\) isomorphic copies of \(IG\).

MSC:

17B01 Identities, free Lie (super)algebras
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References:

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