Derivations and biderivations in dialgebras

Abstract

The concepts of derivations and antiderivations for Leibniz algebras naturally arise from the inner operators determined by their algebraic structure. In this paper, we introduce the corresponding analogues in the setting of dialgebras, which we call diderivations, and examine their structural properties in relation to classical derivations and multiplicative operators. Our approach is based on the study of left and right multiplication operators and on the construction of the Leibniz algebra generated by biderivations, thereby providing a systematic operator-theoretic framework that unifies several derivation-like structures. In addition to the general theory, we present a complete classification of the spaces of diderivations for dialgebras of dimensions two and three, obtained through explicit computations. These low-dimensional results not only illustrate the general constructions but also reveal structural patterns that inform possible extensions to higher dimensions and more intricate algebraic contexts.