Lie isomorphisms of triangular and block-triangular matrix algebras over commutative rings

Date

2016-08-24

Authors

Cecil, Anthony John

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Abstract

For many matrix algebras, every associative automorphism is inner. We discuss results by Đoković that a non-associative Lie automorphism φ of a triangular matrix algebra Tₙ over a connected unital commutative ring, is of the form φ(A)=SAS⁻¹ + τ(A)I or φ(A)=−SJ Aᵀ JS⁻¹ + τ(A)I, where S ∈ Tₙ is invertible, J is an antidiagonal permutation matrix, and τ is a generalized trace. We incorporate additional arguments by Cao that extended Đoković’s result to unital commutative rings containing nontrivial idempotents. Following this we develop new results for Lie isomorphisms of block upper-triangular matrix algebras over unique factorization domains. We build on an approach used by Marcoux and Sourour to characterize Lie isomorphisms of nest algebras over separable Hilbert spaces. We find that these Lie isomorphisms generally follow the form φ = σ + τ where σ is either an associative isomorphism or the negative of an associative anti-isomorphism, and τ is an additive mapping into the center, which maps commutators to zero. This echoes established results by Martindale for simple and prime rings.

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Keywords

Mathematics, Linear Algebra, Matrix Algebras, Ring Theory, Lie Isomorphisms, Triangular Algebras, Block-Triangular Algebras

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