Derivations of Mackey algebras

Abstract

We describe derivations of finitary Mackey algebras over fields of characteristics not equal to $2.$ We prove that an arbitrary derivation of an associative finitary Mackey algebra or one of the Lie algebras $\mathfrak{sl}_{\infty}(V|W)$, $\mathfrak{o}_{\infty}(f)$ is an adjoint operator of an element in the corresponding Mackey algebra. It provides a description of the derivations of all algebras in the Baranov-Strade classification of finitary simple Lie algebras. The proof is based on N. Jacobson's result on derivations of associative algebras of linear transformations of an infinite-dimensional vector space and the results on Herstein's conjectures.