Local and 2-Local Derivations of Finite-Dimensional Zinbiel Algebras

Abstract

We study local and $2$-local derivations of finite-dimensional Zinbiel algebras. Over any field of characteristic zero, every Zinbiel algebra $A$ with $A^2 \neq 0$ and $\dim A \leq 4$ admits a proper local derivation. The proof uses a filtration argument. We exhibit two five-dimensional Zinbiel algebras with nonzero multiplication whose derivations are all central. Every local derivation on either algebra is therefore a derivation, so the minimum dimension for local rigidity with nonzero multiplication is five. Annihilator-valued interpolation gives a general construction of proper local derivations. When $A^3 = 0$, the multiplication tensor describes all derivations and local derivations. For a finite-dimensional nilpotent algebra $A$ over any field, we show that every $2$-local derivation is a derivation if and only if $\dim(A/A^2) \leq 1$. The proof combines the annihilator interpolation of Sh. Ayupov, K. Kudaybergenov and B. Omirov with a one-generator argument. When all derivations are central, every $2$-local derivation is obtained by composing a homogeneous map $A/A^2 \to \text{Ann}(A)$ with the quotient map, and every such composition is a $2$-local derivation. The five-dimensional examples extend to locally rigid Zinbiel algebras with nonlinear $2$-local derivations in every dimension at least five. We also show that derivations given by left multiplication form a Lie ideal of nilpotency class at most two and give sufficient conditions for the existence of left outer derivations. This ideal is zero in our locally rigid examples.