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 $\delta$-derivations. As corollaries, we obtain a complete description of $\delta$-derivations, $\delta$-biderivations, and transposed $\delta$-Poisson structures on $\mathcal{B}(q)$ for $\delta \notin \{0;1\}$. In particular, for $\delta \notin \{0;1\}$, $\mathcal{B}(q)$ up to isomorphism admits only one nontrivial transposed $\delta$-Poisson structure if $q\in \mathbb{Z}$, and $\mathcal{B}(q)$ does not admit nontrivial transposed $\delta$-Poisson structures if $q\notin \mathbb{Z}$.