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Peer-reviewedCC BY 4.0Axioms

n-Derivations of Lie Superalgebras: Structure, Decomposition, and Central Extensions

  • Abu Zaid Ansari1
  • Mukhtar Ahmad2
  • Majed Albaity3
  • Faiza Shujat4
4 affiliations
  1. 1Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
  2. 2Algebra Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha no. 10, Bandung 40132, Indonesia
  3. 3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80348, Jeddah 22254, Saudi Arabia
  4. 4Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
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Abstract

Let n≥2 be fixed, and let Λ be a Lie superalgebra over a commutative ring R. We investigate the structure of the algebra Dern(Λ) of n-derivations. We show that Dern(Λ) is a Z2-graded Lie superalgebra and that its elements preserve n-perfect graded ideals, thereby inducing n-derivations on the corresponding quotients. We obtain a direct-sum decomposition with explicit cross terms and describe n-derivations of split central extensions, including the contributions from maps into the n-center. For perfect centerless Lie superalgebras that are (n−1)-torsion-free, we prove that every n-derivation is an ordinary derivation; consequently, all n-derivations are inner when all ordinary derivations are inner. We also exhibit solvable examples for which Der(Λ) is properly contained in Dern(Λ). These results clarify the roles of the grading, ideals, centers, and split central extensions in the theory of n-derivations.

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Cite

BibTeXRIS
BibTeX
@article{ansari2026nderivations,
  title = {{n-Derivations of Lie Superalgebras: Structure, Decomposition, and Central Extensions}},
  author = {Abu Zaid Ansari and Mukhtar Ahmad and Majed Albaity and Faiza Shujat},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15110794},
  url = {https://www.mdpi.com/2075-1680/15/11/794},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - n-Derivations of Lie Superalgebras: Structure, Decomposition, and Central Extensions
AU  - Abu Zaid Ansari
AU  - Mukhtar Ahmad
AU  - Majed Albaity
AU  - Faiza Shujat
PY  - 2026
DA  - 2026/10/23
JO  - Axioms
DO  - 10.3390/axioms15110794
UR  - https://www.mdpi.com/2075-1680/15/11/794
AB  - Let n≥2 be fixed, and let Λ be a Lie superalgebra over a commutative ring R. We investigate the structure of the algebra Dern(Λ) of n-derivations. We show that Dern(Λ) is a Z2-graded Lie superalgebra and that its elements preserve n-perfect graded ideals, thereby inducing n-derivations on the corresponding quotients. We obtain a direct-sum decomposition with explicit cross terms and describe n-derivations of split central extensions, including the contributions from maps into the n-center. For perfect centerless Lie superalgebras that are (n−1)-torsion-free, we prove that every n-derivation is an ordinary derivation; consequently, all n-derivations are inner when all ordinary derivations are inner. We also exhibit solvable examples for which Der(Λ) is properly contained in Dern(Λ). These results clarify the roles of the grading, ideals, centers, and split central extensions in the theory of n-derivations.
SN  - 2075-1680
ER  -