<p>In this article, we address the characterization of Lie <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-centralizers and Jordan <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-centralizers for an automorphism <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> based on the characterization of Lie centralizers and Jordan centralizers. By applying these results, we provide the characterization of Lie <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-centralizers and Jordan <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-centralizers for any automorphism <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> on generalized matrix algebras and triangular algebra. Thus, we fully answer the question Question 6.1 of [X. Liang, M. Wang and M. Zhang, <i>Centralizers with automorphisms of triangular algebras</i>, Ricerche di Matematica, 2025, https://doi.org/10.1007/s11587-025-00958-w] regarding the relationship between <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-centralizers, Lie <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-centralizers, and Jordan <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-centralizers for any arbitrary automorphism <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> on a triangular algebra, and even extend these characterizations to other algebras.</p>

A note on characterization of $\sigma $-centralizers when $\sigma $ is an automorphism

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发表日期：

2026年2月25日

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Roonak Behfar,
Hoger Ghahramani

摘要

In this article, we address the characterization of Lie $\sigma $ $σ$ -centralizers and Jordan $\sigma $ $σ$ -centralizers for an automorphism $\sigma $ $σ$ based on the characterization of Lie centralizers and Jordan centralizers. By applying these results, we provide the characterization of Lie $\sigma $ $σ$ -centralizers and Jordan $\sigma $ $σ$ -centralizers for any automorphism $\sigma $ $σ$ on generalized matrix algebras and triangular algebra. Thus, we fully answer the question Question 6.1 of [X. Liang, M. Wang and M. Zhang, Centralizers with automorphisms of triangular algebras, Ricerche di Matematica, 2025, https://doi.org/10.1007/s11587-025-00958-w] regarding the relationship between $\sigma $ $σ$ -centralizers, Lie $\sigma $ $σ$ -centralizers, and Jordan $\sigma $ $σ$ -centralizers for any arbitrary automorphism $\sigma $ $σ$ on a triangular algebra, and even extend these characterizations to other algebras.

A note on characterization of \(\sigma \)-centralizers when \(\sigma \) is an automorphism

摘要