Fixed points of automorphisms of free Lie algebras

摘要

1. Introduction and statement of results. If V is a finite-dimensional vector space over a field K then there is a free Lie algebra L over K with the property that Kis a subspace of L and every basis of V is a free generating set of L. We call L the free Lie algebra on V. We may write L as a vector space direct sum L = O Ln where, for each positive. Integer n, Ln is the homogeneous component of degree n, that is, the subspace spanned by all left-normed Lie products [v1,v2,...,v_n] with V1,...,vnV. Clearly each Ln is finite-dimensional. In fact, if V has dimension m then the dimension of Ln is given by Witt's formula: where d ranges over the positive divisors of n and u is the Mobius function (see, for example, [1, ch. 2, ξ3.3]).