Local and 2-local $\frac{1}{2}$-derivations on filiform-type associative algebras

Abstract

This paper is devoted to local and 2-local $\frac{1}{2}$-derivations of null-filiform, filiform, and naturally graded quasi-filiform associative algebras. We obtain explicit descriptions of the corresponding $\frac{1}{2}$-derivations and local $\frac{1}{2}$-derivations and show that non-derivation local maps occur throughout these classes. For 2-local maps, however, a sharp dichotomy appears: every 2-local $\frac{1}{2}$-derivation of a null-filiform associative algebra is a $\frac{1}{2}$-derivation, whereas every filiform and naturally graded quasi-filiform family considered here admits a 2-local $\frac{1}{2}$-derivation which is not a $\frac{1}{2}$-derivation.