A generalization on derivations of Lie algebras

Hongliang Chang; Yin Chen; Runxuan Zhang; Hongliang Chang; Yin Chen; Runxuan Zhang

doi:10.3934/era.2020124

Electronic Research Archive

2021, Volume 29, Issue 3: 2457-2473. doi: 10.3934/era.2020124

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A generalization on derivations of Lie algebras

School of Mathematics and Statistics, Northeast Normal University, Changchun, China

Received: 01 April 2020 Revised: 01 November 2020 Published: 14 December 2020
17B40

We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.
- Generalized derivation,
- automorphism,
- G-derivation
Citation: Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras[J]. Electronic Research Archive, 2021, 29(3): 2457-2473. doi: 10.3934/era.2020124

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Abstract

We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.

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