NECESSARY CONDITION FOR $IA$-AUTOMORPHISMS IN LEIBNIZ ALGEBRAS

Abstract

Let $F$ be the free Leibniz algebra generated by the set $% X=\{x_{1},...,x_{n}\}$ over the field $K$ of characteristic $0$. consider $R$ as an ideal of $F$. This study initially derives an explicit matrix representation for the $IA$-automorphisms of the Leibniz algebra $F/R^{\prime }$. Subsequently, we establish a necessary condition for an $IA$% -endomorphism of $F/R^{\prime }$ to be an $IA$-automorphism. This method is explicitly based on Dieudonn\'{e} determinant.