Derivations in Dialgebras Derivations and Biderivations in Dialgebras

TL;DR

This paper introduces and studies diderivations in dialgebras, generalizing derivations from Leibniz algebras, and provides a classification for low-dimensional cases.

Contribution

It defines diderivations for dialgebras, explores their properties, and classifies these operators in dimensions two and three.

Findings

Complete classification of diderivations in 2D and 3D dialgebras.

Unified framework for derivation-like operators in dialgebras.

Structural patterns identified in low-dimensional cases.

Abstract

The concepts of derivations and right derivations for Leibniz algebras and $K$-B quasi-Jordan algebras naturally arise from the inner derivations determined by their algebraic structures. In this paper we introduce the corresponding analogues for dialgebras, which we call diderivations, and examine their properties in relation to antiderivations and right derivations. Our approach is based on the study of multiplicative operators and on the construction of the Leibniz algebra generated by biderivations, thereby providing a systematic framework that unifies several types of derivation-like operators. 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 exemplify the general constructions but also reveal structural patterns that inform possible extensions to higher dimensions and more intricate algebraic contexts.28