The image of Lie polynomials on real Lie algebras of dimension up to 3

Abstract

Let Fn be the free Lie algebra over R of rank n generated by y1,…,yn, and let f∈Fn′ be a multilinear Lie polynomial contained in the commutator ideal Fn′ of Fn. In this paper, we determine the imageImf={f(w1,…,wn)|wi∈L,i=1,…,n}⊂L, for Lie algebras L of dimension ≤3, and of the Lie algebra of dimension 4 stated in a paper of Baker dating back to 1901. In all the cases studied, the L'vov-Kaplansky Conjecture has a positive answer.