Decomposition of Jordan automorphisms of strictly triangular matrix algebra over local rings
Let \(R\) be a commutative ring with identity 1 and \(A\) an algebra over \(R\). A bijective \(R\)-linear map \(f: A \to A\) is called a Jordan automorphism if \(f(ab+ba) = f(a)f(b)+f(b)f(a)\) for all \(a,b \in A\). The study of Jordan isomorphisms of associative algebras has been developed during the last half-century. Clearly, automorphisms and anti-automorphisms are examples of Jordan automorphisms. \textit{K. I. Beĭdar, M. Brešar} and \textit{M. A. Chebotar} [Linear Algebra Appl. 312, 197--201 (2000; Zbl 0962.15007)] gave an example of a Jordan automorphism which is neither an automorphism nor an anti-automorphism. Many papers are concerned with the study of automorphisms and Lie automorphisms of \(T_n(R)\), the algebra of all triangular matrices over \(R\). \textit{Y. A. Cao} [Linear Algebra Appl. 329, 175--187 (2001; Zbl 0987.17010)] studied the automorphisms of the nilpotent Lie algebra of all strictly upper triangular \((n+1)\times (n+1)\) matrices over a local ring with 2 as unit or an integral domain of characteristic not 2, and gave a decomposition for the above isomorphisms. In the present paper, the authors consider an analogous problem concerning Jordan automorphisms of \(N_{n+1}(R)\), the algebra of all strictly upper triangular \((n+1)\times (n+1)\) matrices over \(R\), where \(R\) is a 2-torsionfree local ring. They prove that any Jordan automorphism of \(N_{n+1}(R)\) can be uniquely expressed as a product of a graph automorphism, a diagonal automorphism, an inner automorphism and a central automorphism for \(n \geq 3\) and \(R\) is a 2-torsionfree local ring. In the case where \(n=1\), any Jordan automorphism of \(N_2(R)\) is a diagonal automorphism. If \(n=2\), any Jordan automorphism of \(N_3(R)\) can be written as a product of graph, diagonal and inner automorphisms.
scientific article; zbMATH DE number 2119134
scientific article