On Hersteinʹs Lie Map Conjectures, II

The theory of functional identities is used to study derivations of Lie algebras arising from associative algebras. Definitive results are obtained modulo algebras of “low dimension.” In particular, Lie derivations of [ , ]/([ , ] ∩ ), where is the Lie algebra of skew elements of a prime algebra with involution and is its center, are described. This solves the last remaining open problem of Herstein on Lie derivations. For a simple algebra with involution the Lie algebra of all derivations of [ , ]/([ , ] ∩ ) is thoroughly analyzed. Maps that act as derivations on arbitrary fixed polynomials are also discussed, and in particular a solution is given for Hersteinʹs question concerning maps of which act like a derivation on xm, m being a fixed odd integer.