On $(\lambda,\mu,\gamma)$-derivations of BiHom-Lie algebras

TL;DR

This paper extends the theory of derivations to BiHom-Lie algebras, classifies generalized derivations for specific parameters, and provides detailed classifications for 2-dimensional cases.

Contribution

It introduces a parameterized definition of generalized derivations for BiHom-Lie algebras and classifies these derivations for Heisenberg and 2-dimensional cases.

Findings

Classification of generalized derivations for Heisenberg BiHom-Lie algebras.

Explicit descriptions of derivations and centroids for 2-dimensional BiHom-Lie algebras.

Identification of classical derivations as a special case with parameters (1,1,1).

Abstract

In this paper, we generalize the results about generalized derivations of Lie algebras to the case of BiHom-Lie algebras. In particular we give the classification of generalized derivations of Heisenberg BiHom-Lie algebras. The definition of the generalized derivation depends on some parameters $ (\lambda,\mu,\gamma)\in \mathbb{C}^{3}. $ In particular for $(\lambda,\mu,\gamma)=(1,1,1)$, we obtain classical concept of derivation of BiHom-Lie algebra and for $(\lambda,\mu,\gamma)=(1,1,0) $ we obtain the centroid of BiHom-Lie algebra. We give classifications of $ 2 $-dimensional BiHom-Lie algebra, centroides and derivations of $ 2 $-dimensional BiHom-Lie algebras.