On generalized derivations of polynomial vector fields Lie algebras (Q6588442)

The author studied Lie algebras \(\mathfrak{P}\) of polynomial vector fields on \( \mathbb R^n \) (\(R\) is the field of real numbers) containing constant vector fields \(\frac{\partial}{\partial x_i}\), \(i=1, \ldots , n\) and the Euler vector field \(E=\sum _{i=1}^n x_i\frac{\partial}{\partial x_i}. \) It is proved in the first part of the paper that the centroid \(C(\mathfrak{P})\) of a Lie algebra \(\mathfrak{P}\) coincides with the quasicentroid \(QC(\mathfrak{P}) \) (recall that, for a Lie algebra \(L\), the centroid \(C(L)\) consists of all the endomorphisms \(f\) on \(L\) such that \(f[X, Y]=[f(X), Y]=[X, f(Y)]\) and the quasicenter \(QC(L)\) is a Lie algebra of all the endomorphisms \(g\) on \(L\) with \([g(X), Y]=[X, g(Y)]\) for all \(X, Y \in L \) ). In the second part of the paper, quasiderivations on Lie algebras \(\mathfrak{P}\) are studied, i.e., endomorphisms on \(\mathfrak{P}\) such that \([f(X), Y]+[X, f(Y)]=g[X, Y]\) for some endomorphism \(g\) and for all \(X, Y \in L. \) The last part of the work is devoted to a description of generalized Lie derivations on \(\mathfrak{P}.\)