ON CENTRAL AUTOMORPHISMS OF FREE METABELIAN LIE ALGEBRAS

Year 2022, Volume: 5 Issue: 2, 61 - 67, 31.07.2022

Başak Erginkara , Şehmus Fındık

https://doi.org/10.33773/jum.1141787

Abstract

Let $F_m$ be the free metabelian Lie algebra of rank $m$ over a field $K$ of characteristic 0. An automorphism $\varphi$ of $F_m$ is called central if $\varphi$
commutes with every inner automorphism of $F_m$. Such automorphisms form the centralizer $\text{\rm C}(\text{\rm Inn}(F_m))$
of inner automorphism group $\text{\rm Inn}(F_m)$ of $F_m$ in $\text{\rm Aut}(F_m)$. We provide an elementary proof to show that $\text{\rm C}(\text{\rm Inn}(F_m))=\text{\rm Inn}(F_m)$.

Keywords

Automorphism, inner, Lie algebras

References

• Yu.A. Bahturin, Identical Relations in Lie Algebras (Russian), Nauka, Moscow, (1985). Translation: VNU Science Press, Utrecht, (1987).
• R.M. Bryant, V. Drensky, Dense subgroups of the automorphism groups of free algebras, Canad. J. Math. 45, pp. 1135-1154 (1993).
• M. J. Curran, D. J. McCaughan, Central automorphisms that are almost inner, Commun. Alg. 29(5), pp. 2081-2087 (2001).
• G. A. Miller, Dense subgroups of the automorphism groups of free algebras, Mess. of Math. 43, pp. 124 (1913-1914).
• A.L. Shmel'kin, Wreath products of Lie algebras and their application in the theory of groups (Russian), Trudy Moskov. Mat. Obshch. 29, pp. 247-260 (1973). Translation: Trans. Moscow Math. Soc. 29, pp. 239-252 (1973).