Let L(V) be the free Lie algebra on a finite‐dimensional vector space V over a field K, with homogeneous components Ln(V) for n ⩾ 1. If G is a group and V is a KG‐module, the action of G extends naturally to L(V), and the Ln(V) become finite‐dimensional KG‐modules, called the Lie powers of V. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable KG‐modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where G has prime order p, and K has characteristic p. As is well known, there are p indecomposables, denoted here by J1,…,Jp, where Jr has dimension r. A theory is developed which provides information about the overall module structure of LV) and gives a recursive method for finding the multiplicities of J1,…,Jp in the Lie powers Ln(V). For example, the theory yields decompositions of L(V) as a direct sum of modules isomorphic either to J1 or to an infinite sum of the form Jr ⊕ J{p‐1} ⊕ J{p‐1} ⊕ … with r ⩾ 2. Closed formulae are obtained for the multiplicities of J1,…, Jp in Ln(Jp and Ln(J{p‐1). For r < p‐1, the indecomposables which occur with non‐zero multiplicity in Ln(Jr) are identified for all sufficiently large n. 2000 Mathematical Subject Classification: 17B01, 20C20.
Abstract. We study the free centre-by-metabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous and fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian 2-group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous and fine homogeneous components of the free centre-by-metabelian Lie algebra over fields of characteristic other than 2. For that we exploit a 6-term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field. Our results correct a partly erroneous theorem in the literature.
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