On conservative algebras of 2-dimensional Algebras
Hassan Oubba
TL;DR
This work analyzes conservative and terminal algebras in the 2‑dimensional setting, focusing on $\mathcal{W}(2)$, $\mathcal{W}_2$, and $\mathcal{S}_2$. It provides a complete description of $\tfrac{1}{2}$-derivations, showing they are always scalar multiplications, and proves that local and $2$-local $\tfrac{1}{2}$-derivations coincide with $\tfrac{1}{2}$-derivations across these algebras. The authors also show that all biderivations vanish: $BDer(\mathcal{W}(2))=BDer(\mathcal{W}_2)=BDer(\mathcal{S}_2)=\{0\}$, including both symmetric and skew-symmetric cases. These results clarify the derivation-like structure of low‑dimensional conservative/terminal algebras and contribute to the understanding of local and 2-local operator theory in nonassociative settings, with explicit descriptions tied to the algebras $\mathcal{W}(2)$, $\mathcal{W}_2$, and $\mathcal{S}_2$.
Abstract
In 1990 Kantor introduced the conservative algebra $\mathcal{W}(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. In case $n >1$ the algebra $\mathcal{W}(n)$ does not belong to well known classes of algebras (such as associative, Lie, Jordan, Leibniz algebras). We describe $\frac{1}{2}$derivations, local (resp. $2$-local) $\frac{1}{2}$-derivations and biderivations of $\mathcal{W}(2)$. We also study similar problems for the algebra $\mathcal{W}_2$ of all commutative algebras on the two-dimensional vector space and the algebra $\mathcal{S}_2$ of all commutative algebras with trace zero multiplication on the two-dimensional space.