ON THE AUTOMORPHISM GROUP OF A FREE METABELIAN LIE ALGEBRA

Abstract

Let K be a field of any characteristic. We prove that a free metabelian Lie algebra M3of rank 3 over K admits wild automorphisms. Moreover, the subgroup I Aut M3of all automorphisms identical modulo the derived subalgebra [Formula: see text] cannot be generated by any finite set of IA-automorphisms together with the sets of all inner and all tame IA-automorphisms. In the case if K is finite the group Aut M3cannot be generated by any finite set of automorphisms together with the sets of all tame, all inner automorphisms and all one-row automorphisms. We present an infinite set of wild IA-automorphisms of M3which generates a free subgroup F∞modulo normal subgroup generated by all tame, all inner and all one-row automorphisms of M3.