Decomposition of Jordan automorphisms of strictly triangular matrix algebra over local rings
type: Item
instance of: scholarly article
title: Decomposition of Jordan automorphisms of strictly triangular matrix algebra over local rings
zbMATH Open document ID: 1056.17025
author: Hong You
author: Xing Tao Wang
published in: Linear Algebra and its Applications
publication date: 2004-11-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17C30
Mathematics Subject Classification ID: 17C50
zbMATH DE Number: 2119134
zbMATH Keywords: Jordan automorphism
zbMATH Keywords: strictly triangular matrix algebra
zbMATH Keywords: local ring
reviewed by: María Concepción López-Díaz
MaRDI profile type: Publication
full work available at URL: j.laa.2004.06.009
OpenAlex ID: W2043056126
cites work: On semi-automorphisms of division algebras
cites work: Jordan homomorphisms of semiprime rings
cites work: Jordan isomorphisms of triangular matrix algebras over a connected commutative ring
cites work: Jordan mappings of semiprime rings
cites work: Automorphisms of certain Lie algebras of upper triangular matrices over a commutative ring
cites work: Automorphisms of the Lie algebra of strictly upper triangular matrices over certain commutative rings
cites work: Jordan Homomorphisms
cites work: Automorphisms of upper triangular matrix rings
cites work: The automorphism group of certain radical matrix rings
cites work: Modular automorphisms preserving idempotence and Jordan isomorphisms of triangular matrices over commutative rings
DOI: 10.1016/J.LAA.2004.06.009
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description: scientific article; zbMATH DE number 2119134