Inner Automorphisms of Lie Algebras Related with Generic 2 x 2 Matrices

TL;DR

This paper investigates the structure of inner automorphisms in Lie algebras generated by generic 2x2 matrices, providing explicit descriptions and extending classical results to new algebraic contexts.

Contribution

It offers a detailed description of inner automorphisms for Lie algebras related to generic 2x2 matrices, extending Baker's classical results to the setting of relatively free algebras.

Findings

Complete description of inner automorphisms for the algebra generated by two generic traceless 2x2 matrices

Explicit multiplication rule for automorphisms in the completed algebra setting

Extension of results to automorphisms of certain nilpotent algebra quotients

Abstract

Let F_m=F_m(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 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 x 2 matrices. We give a complete description of the group of inner automorphisms of the completion of L. As a consequence we obtain similar results for the automorphisms of the relatively free algebra F_m/F_m^{c+1} in the subvariety of var(sl(2,K)) consisting of all nilpotent algebras of class at most c in var(sl(2,K)).