Inner automorphisms of Lie algebras related with generic 2 × 2 matrices

Abstract

Let Fm = Fm(var(sl₂(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl₂(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 Fmˆ of Fm with respect to the formal power series topology. Our results are more precise for m = 2 when F₂ is isomorphic to the Lie algebra L generated by two generic traceless 2×2 matrices. We give a complete description of the group of inner automorphisms of Lˆ. As a consequence we obtain similar results for the automorphisms of the relatively free algebra Fm / Fm c⁺¹ = Fm(var(sl₂(K)) ∩ Nc)