Local automorphisms of $n$-dimensional naturally graded quasi-filiform Leibniz algebra of type I

The notions of a local automorphism for Lie algebras are defined as similar to the associative case. Every automorphism of a Lie algebra $\mathcal{L}$ is a local automorphism. For a given Lie algebra $\mathcal{L}$, the main problem concerning these notions is to prove that they automatically become an automorphism or to give examples of local automorphisms of $\mathcal{L}$, which are not automorphisms. In this paper, we study local automorphisms on quasi-filiform Leibniz algebras. It is proved that quasi-filiform Leibniz algebras of type I, as a rule, admit local automorphisms which are not automorphisms.