Quasi-derivations of Witt and related algebras

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 $$\frac{1}{2}$$ 1 2 -derivation; a similar result is obtained for the Virasoro algebra. A different situation appears for Lie algebras $$\mathscr {W}(a,b):$$ W ( a , b ) : In the case of $$b=-1,$$ b = - 1 , they do not have interesting examples of quasi-derivations, but the case of $$b\ne -1$$ b ≠ - 1 provides some new non-trivial examples of quasi-derivations. We also completely describe all quasi-derivations of $$\mathscr {W}(a,b).$$ 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 $$\delta $$ δ -derivations and transposed $$\delta $$ δ -Poisson structures on cited Lie algebras. In particular, we proved that each $$\mathscr {W}(a,b)$$ W ( a , b ) admits a non-trivial transposed $$\frac{1}{1-b}$$ 1 1 - b -Poisson structure.