Quasi-derivations of Witt and related algebras

TL;DR

This paper characterizes quasi-derivations of the Witt algebra and related algebras, revealing their structure and providing explicit descriptions, including for Novikov-Witt algebras, with implications for Poisson structures.

Contribution

It explicitly describes all quasi-derivations of Witt and related algebras, extending understanding of their algebraic structures and Poisson geometry.

Findings

Quasi-derivations of Witt algebra are sums of derivations and 1/2-derivations.

Complete description of quasi-derivations for ${ m W}(a,b)$ algebras.

Existence of nontrivial transposed 1/(1-b)-Poisson structures on ${ m W}(a,b)$.

Abstract

In the present work, we compute quasi-derivations of the Witt algebra and some algebras well-related to the Witt algebra. Namely, we prove that each quasi-derivation of the Witt algebra is a sum of a derivation and a $\frac{1}{2}$-derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras ${\mathcal W}(a,b):$ in the case of $b=-1,$ they do not have interesting examples of quasi-derivations, but the case of $b\neq-1$ provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of ${\mathcal W}(a,b).$ As a corollary, we describe the derivations and quasi-derivations of the Novikov-Witt and admissible Novikov-Witt algebras previously constructed by Bai and his co-authors; and $\delta$-derivations and transposed $\delta$-Poisson structures on cited Lie algebras. In particular, we proved that each ${\mathcal W}(a,b)$ admits a nontrivial transposed $\frac 1{1-b}$-Poisson structure.