Quasi-derivations of Block Lie algebras $\mathcal{B}(q)$

Abstract

We compute quasi-derivations of Block Lie algebras $\mathcal{B}(q)$. In particular, we prove that $\mathcal{B}(q)$ has various nontrivial quasi-derivations, i.e., quasi-derivations different from derivations and $δ$-derivations. As corollaries, we obtain a complete description of $δ$-derivations, $δ$-biderivations, and transposed $δ$-Poisson structures on $\mathcal{B}(q)$ for $δ\notin\{0;1\}$. In particular, for $δ\notin\{0;1\}$, $\mathcal{B}(q)$ up to isomorphism admits only one nontrivial transposed $δ$-Poisson structure if $q\in\mathbb Z$, and $\mathcal{B}(q)$ does not admit nontrivial transposed $δ$-Poisson structures if $q\notin\mathbb Z$.