1Let L be a free Lie algebra of finite rank over a field K and let X be a free generating set of L. Then L is a graded algebra,where L n is the K-span of all left-normed Lie productsLet g L → L be a graded algebra automorphism of order 2. If 2 is invertible in K then L has a homogeneous basis consisting of eigenvectors of g with eigenvalues ±1. For example, every Hall basis constructed from a free generating set consisting of eigenvectors has this property. Here we are concerned with the case where K is of characteristic 2. Then L has a homogeneous basis which is invariant under the action of g. In this paper we give an explicit construction of such a basis in terms of a g-invariant free generating set X. In fact, we first construct a g-invariant basis * for the free restricted Lie algebra on X (Theorem 1), and then we get as a subset of that basis (Theorem 2).The motivation for this work came from a problem posed by Bryant [2]. If the field K is replaced by the ring of integers , that is L is the free Lie ring on X, and g is a graded algebra automorphism of order 2, then L has a -basis such that ∪ − is a g-invariant set. Bryant asked if one could actually find a basis with this high degree of symmetry (Problem D in [2]). Theorem 2 solves this problem modulo 2.Our construction yields a new proof of results from [3,7] about the structure of L (with K a field of characteristic 2) as a module for the cyclic group of order 2 acting on L via a graded algebra automorphism. We mention that in the case where K = GF 2 and X = 2, the results of [3] have been exploited in [6] to determine the multiplicities of the indecomposable GL 2 2 -modules (induced from the natural action of GL 2 2 337
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