Central derivations of low-dimensional Zinbiel algebras

The study of central derivations in low-dimensional algebraic structures is a crucial area of research in mathematics, with applications in understanding the internal symmetries and deformations of these structures. In this article, we investigate the central derivations of complex Zinbiel algebras of dimension $\le 4$. Key properties of the central derivation algebras are presented, including their structures and dimensions. The results are summarized in a tabular format, providing a clear classification of decomposable and indecomposable centroids based on these derivations. Specifically, we show that the centroid of two-dimensional Zinbiel algebras is indecomposable, while in three-dimensional Zinbiel algebras, centroids such as ${\text{A}}_3^3$, ${\text{A}}_4^3$, ${\text{A}}_6^3$, and ${\text{A}}_7^3$ are decomposable. For four-dimensional Zinbiel algebras, centroids including ${\text{A}}_1^4$, ${\text{A}}_3^4$, ${\text{A}}_5^4$, ${\text{A}}_9^4$, ${\text{A}}_{10}^4$, ${\text{A}}_{11}^4$, and ${\text{A}}_{16}^4$ are decomposable. Furthermore, the dimensions of central derivation algebras vary across different dimensions: two-dimensional Zinbiel algebras have central derivation dimensions of one, while in three-dimensional and four-dimensional cases, these dimensions range from zero to nine.