Additivity of Lie multiplicative maps on triangular algebras

摘要

Let and be unital algebras over a commutative ring , and be a (, )-bimodule, which is faithful as a left -module and also as a right -module. Let  = Tri(, , ) be the triangular algebra and any algebra over . Assume that Φ :  →  is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST − TS) = Φ(S)Φ(T) − Φ(T)Φ(S) for all S, T  . Then Φ(S + T) = Φ(S) + Φ(T) + Z _S,T for all S, T  , where Z _S,T is an element in the centre () of depending on S and T.