ON CENTRAL AUTOMORPHISMS OF FREE METABELIAN LIE ALGEBRAS

Yıl 2022, Cilt: 5 Sayı: 2, 61 - 67, 31.07.2022

Başak Erginkara , Şehmus Fındık

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

Öz

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 $\text{\rm C}(\text{\rm Inn}(F_m))=\text{\rm Inn}(F_m)$.

Anahtar Kelimeler

Automorphism, inner, Lie algebras

Kaynakça

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• 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).