Additivity of Lie multiplicative maps on triangular algebras

Abstract

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