Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras
A linear map \(\varphi\) on an algebra is said to be commutativity preserving in both directions when the condition \(ab=ba\) holds if and only if \(\varphi(a)\varphi(b)=\varphi(b)\varphi(a)\). In this paper, the algebra \(\mathcal T_n\) of upper triangular \(n\times n\) matrices over a field is considered. It is shown that for \(n>2\) the commutativity preserving linear map in both directions \(\varphi: \mathcal T_n\rightarrow\mathcal T_n\) is either the sum of an algebra automorphism and a map into the center, or the sum of the negative of an algebra anti-automorphism and a map into the center. That is, \(\varphi(T)=cS^{-1}TS+f(T)I\) or \(\varphi(T)=cS^{-1}JT ^t JS+f(T)I\), where \(c\) is a non-zero scalar, \(S\in\mathcal T_n\) is invertible, \(f(T)\) is a linear functional, \(J=(\delta_{i,n+1-i})\). This result is false for \(n=2\). As a consequence, \(\varphi\) is a Lie automorphism if and only if \(\varphi(T)=S^{-1}TS+\tau(T)I\) or \(\varphi(T)=-S^{-1}JT ^t J S+\tau(T)I\), where \(\tau\) is a linear functional satisfying \(\tau(AB-BA)=0\) for every \(A,B\in\mathcal T_n\) and \(\tau(I)\neq-1\).
scientific article; zbMATH DE number 1331336