Derivations, Centroids and Iner-Derivations of $\textbf{5}$-Dimensional Nilpotent Complex Associative Algebras

TL;DR

This work addresses the problem of understanding the symmetry structures of $5$-dimensional complex nilpotent associative algebras by computing the derivation spaces $\mathrm{Der}(A)$, centroids $\mathrm{Cent}(A)$, and inner derivations $\mathrm{Inn}(A)$. Building on a classification of lower-dimensional nilpotent algebras, the authors present $31$ representative algebras $A_i$ with explicit multiplication tables and centers, and derive the corresponding linear systems to obtain $\mathrm{Der}(A_i)$, $\mathrm{Cent}(A_i)$, and $\mathrm{Inn}(A_i)$, including their dimensions. They establish the universal bounds $\dim(\mathrm{Inn}(A))\le\dim(\mathrm{Cent}(A))\le\dim(\mathrm{Der}(A))$ and report dimension ranges for each invariant (Der: $2$–$9$, Cent: $2$–$8$, Inn: $2$–$4$). The resulting computational atlas advances understanding of rigidity and deformation theory for small nilpotent associative algebras and provides a foundation for geometric approaches to varieties of nilpotent algebras.

Abstract

This study focuses on the analysis of derivations, centroids, and inner derivations of 5-dimensional complex nilpotent associative algebras. It presents the classification of these algebras of dimension less than five, as well as the classifications of their corresponding derivations, centroids and iner-derivations.

Theorems & Definitions (8)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Proposition 3.1
• Proposition 4.1
• Proposition 5.1
• Corollary 5.1
• Remark 5.1