On Herstein's 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 module algebras of "low dimension." In particular, Lie derivations of [K,K]/([K,K] boolean AND E), where K is the Lie algebra of skew elements of a prime algebra with involution and E 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 [K,K]/([K,K] boolean AND E) is thoroughly analyzed. Maps that act as derivations on arbitrary fixed polynomials are also discussed, and in particular a solution is given for Herstein's question concerning maps of K which act like a derivation on x(m), m being a fixed odd integer. (C) 2001 Academic Press.