Computational Approaches to Derivations and Automorphism Groups of Associative Algebras
Ahmed Zahari Abdou, Bouzid Mosbahi
TL;DR
The paper addresses deriving derivations and automorphism groups for finite-dimensional associative algebras over $\mathbb{C}$ by combining classification results with computer algebra computations. It formulates an algorithmic pipeline for derivations based on structure constants $\gamma^k_{ij}$ and the Leibniz rule, and for automorphisms via polynomial constraints from $f(e_i\cdot e_j)=f(e_i)\cdot f(e_j)$, enabling explicit descriptions of $Der(A)$ and $Aut(A)$. The authors provide explicit parametrizations and group structures for all algebras of dimensions $2$, $3$, and $4$, and report the dimensional ranges of $Der(A)$ and $Aut(A)$ (e.g., $\dim Der$ in $[0,2]$ for $2$-D, $[2,4]$ for $3$-D, $[0,12]$ for $4$-D, and $\dim Aut$ in $[1,12]$). These results enhance the understanding of symmetries and geometric classification of low-dimensional associative algebras and demonstrate the utility of computer algebra tools in structural algebra.
Abstract
This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with computational tools like Mathematica and Maple, we offer detailed descriptions of the derivations and automorphism groups for these algebras. Our analysis of these groups helps to uncover important structural features and symmetries in low-dimensional associative algebras.