2021/11/30 by Brešar, Matej
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2111.15237
Let A be a finite-dimensional algebra over a field F with char(F)≠ 2. We show that a linear map D:A→ A satisfying xD(x)x∈ [A,A] for every x∈ A is the sum of an inner derivation and a linear map whose image lies in the radical of A. Assuming additionally that A is semisimple and char(F)≠ 3, we show that a linear map T:A→ A satisfies T(x)3- x3 ∈ [A,A] for every x∈ A if and only if there exist a Jordan automorphism J of A lying in the multiplication algebra of A and a central element α satisfying α3=1 such that T(x)=αJ(x) for all x∈ A. These two results are applied to the study of local derivations and local (Jordan) automorphisms. In particular, the second result is used to prove that every local Jordan automorphism of a finite-dimensional simple algebra A (over a field F with char(F)≠ 2,3) is a Jordan automorphism.