Automorphisms of Free Nilpotent Lie Algebras

Abstract

Let Fm be the free Lie algebra of rank m over a field K of characteristic 0 freely generated by the set ﹛x1,… ,xm﹜, m ≧ 2. Cohn [7] proved that the automorphism group Aut Fm of the K-algebra Fm is generated by the following automorphisms: (i) automorphisms which are induced by the action of the general linear group GLm (= GLm(K)) on the subspace of Fm spanned by ﹛x1, … ,xm﹜; (ii) automorphisms of the form x1 → x1 +f(x2,… ,xm),Xk → xk, k ≠ 1, where the polynomial f(x2,…,xm) does not depend on x1.