Extended derivations of algebras

Abstract

We study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of derivations by considering a suitable notion of equivalence when we restrict attention to such algebras. Afterwards, we investigate a link between the well-known $(\alpha,\beta,\gamma)$-derivations and some new extended derivations obtained in the above-mentioned classification. Finally, we present applications of extended derivations to study degenerations of algebras.