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Biderivations of Lie algebras

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  08 January 2025

Qiufan Chen ,
Yufeng Yao Open the ORCID record for Yufeng Yao [Opens in a new window]  and
Kaiming Zhao Open the ORCID record for Kaiming Zhao [Opens in a new window]
Qiufan Chen
Affiliation:
Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China
Yufeng Yao*
Affiliation:
Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China
Kaiming Zhao
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5, and School of Mathematical Science, Hebei Normal (Teachers) University, Shijiazhuang, Hebei, 050024, China e-mail: kzhao@wlu.ca
*
e-mail: yfyao@shmtu.edu.cn
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Abstract

In this paper, we first introduce the concept of symmetric biderivation radicals and characteristic subalgebras of Lie algebras and study their properties. Based on these results, we precisely determine biderivations of some Lie algebras including finite-dimensional simple Lie algebras over arbitrary fields of characteristic not $2$ or $3$, and the Witt algebras $\mathcal {W}^+_n$ over fields of characteristic $0$. As an application, commutative post-Lie algebra structure on the aforementioned Lie algebras is shown to be trivial.

Keywords

Symmetric biderivationWitt algebrafinite-dimensional simple Lie algebrasymmetric radicalcharacteristic subalgebra

MSC classification

Primary: 17B20: Simple, semisimple, reductive (super)algebras 17B40: Automorphisms, derivations, other operators 17B65: Infinite-dimensional Lie (super)algebras 17B66: Lie algebras of vector fields and related (super) algebras

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 2 , June 2025 , pp. 440 - 450
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12271345 and 12071136) and NSERC (311907-2020).

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