Inner and outer automorphisms of relatively free algebras (Q2844631)

Let \(K\) be a field of zero characteristic, \(L _ m\) the free Lie algebra over \(K\) of rank \(m \geq 2\), and for an arbitrary Lie \(K\)-algebra, let \(I(G) = I _ m(G)\) be the ideal of \(L _ m\) consisting of all Lie polynomial identities in \(G\) in \(m\) variables. As usual, the quotient \(K\)-algebra \(F _ m(G) = F _ m(\text{var}\, G) = L _ m/I(G)\) is called a relatively free \(K\)-algebra of rank \(m\) in the variety of Lie algebras generated by \(G\). The paper under review is devoted to the study of \(K\)-automorphisms of \(F _ m(G)\) in case \(F _ m(G)\) is a nilpotent algebra. It describes the groups of inner and of outer automorphisms of the free metabelian and nilpotent of class \(c\) \(K\)-algebra \(L _ m/(L _ m ^ {''} + L _ m ^ {c+1})\) as well as the group of inner automorphisms of the relatively free algebra of rank \(2\) in the variety \(\text{var}\,sl_ 2(K) \cap N _ c\), where \(N _ c\) is the variety of nilpotent \(K\)-algebras of nilpotency class \(\leq c\). For that purpose, the authors first describe the group of inner automorphism of the completion of the relatively free Lie \(K\)-algebras \(L _ m/L _ m ^ {''}\) and \(F _ 2(\text{var}\, sl_ 2(K))\) with respect to the formal power series topology. When \(F _ {m}\) is the free metabelian Lie algebra of rank \(m\), they also describe the group of continuous outer automorphisms of the completion \(\widehat F _ m\).