Generalized derivations and quasi derivations of current $ \text{H} $om-$ \text{L} $ie algebras

Omaima Alshanqiti; Ashutosh Pandey; Mani Shankar Pandey; Omaima Alshanqiti; Ashutosh Pandey; Mani Shankar Pandey

doi:10.3934/math.2026199

AIMS Mathematics

2026, Volume 11, Issue 2: 4872-4889. doi: 10.3934/math.2026199

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Generalized derivations and quasi derivations of current $ \text{H} $om-$ \text{L} $ie algebras

1.
Umm Al-Qura University, Makkah, Saudi Arabia
2.
School of Basic Sciences, Department of Mathematics, Indian Institute of Technology Bhubaneshwar, India
3.
Department of Sciences, Indian Institute of Information Technology Design and Manufacturing Kurnool, Andhra Pradesh 518008, India

Received: 01 December 2025 Revised: 02 February 2026 Accepted: 10 February 2026 Published: 27 February 2026
MSC : 17B40, 16W25, 17B99

In this paper, we investigate the structure of generalized derivations and quasiderivations of the tensor product algebra $ \mathcal{H} \otimes \mathcal{A} $, where $ \mathcal{H} $ is a Hom-Lie algebra and $ \mathcal{A} $ is a commutative associative algebra with unity over the field $ \mathcal{K} $. We examine whether every generalized derivation (respectively, quasiderivation) of $ \mathcal{H} \otimes \mathcal{A} $ can be expressed as a sum of a derivation and a linear map in the centroid of $ \mathcal{H} \otimes \mathcal{A} $, provided that this property holds for $ \mathcal{H} $. We establish a partial affirmative result under suitable conditions and discuss the algebraic implications of our findings.
- Hom-Lie algebra,
- generalized derivations,
- quasiderivation,
- centroid
Citation: Omaima Alshanqiti, Ashutosh Pandey, Mani Shankar Pandey. Generalized derivations and quasi derivations of current $ \text{H} $om-$ \text{L} $ie algebras[J]. AIMS Mathematics, 2026, 11(2): 4872-4889. doi: 10.3934/math.2026199

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Abstract

In this paper, we investigate the structure of generalized derivations and quasiderivations of the tensor product algebra $ \mathcal{H} \otimes \mathcal{A} $, where $ \mathcal{H} $ is a Hom-Lie algebra and $ \mathcal{A} $ is a commutative associative algebra with unity over the field $ \mathcal{K} $. We examine whether every generalized derivation (respectively, quasiderivation) of $ \mathcal{H} \otimes \mathcal{A} $ can be expressed as a sum of a derivation and a linear map in the centroid of $ \mathcal{H} \otimes \mathcal{A} $, provided that this property holds for $ \mathcal{H} $. We establish a partial affirmative result under suitable conditions and discuss the algebraic implications of our findings.

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