Inner automorphisms of Lie algebras related with generic \(2\times 2\) matrices

Abstract

Let \(F_m=F_m(\text{var}(sl_2(K)))\) be the relatively free algebra of rank \(m\) in the variety of Lie algebras generated by the algebra \(sl_2(K)\) over a field \(K\) of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion \(\widehat{F_m}\) of \(F_m\) with respect to the formal power series topology. Our results are more precise for \(m=2\) when \(F_2\) is isomorphic to the Lie algebra \(L\) generated by two generic traceless \(2\times 2\) matrices. We give a complete description of the group of inner automorphisms of \(\widehat L\). As a consequence we obtain similar results for the automorphisms of the relatively free algebra \(F_m/F_m^{c+1}=F_m(\text{var}(sl_2(K))\cap {\mathfrak N}_c)\) in the subvariety of \(\text{var}(sl_2(K))\) consisting of all nilpotent algebras of class at most \(c\) in \(\text{var}(sl_2(K))\).