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Kupershmidt--Nijenhuis structures on pre-Leibniz algebras

  • Sami Benabdelhafidh
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Abstract

In this paper, we first construct the dual of a representation of a pre-Leibniz algebra and study infinitesimal deformations of a pre-Leibniz algebra with a representation, which lead to Nijenhuis pairs, characterizing trivial deformations, and to dual-Nijenhuis pairs. Building on these notions, we introduce Kupershmidt-(dual-)Nijenhuis structures and show that they give rise to hierarchies of pairwise compatible Kupershmidt operators, the converse holding under an invertibility condition. Finally, we show that Rota--Baxter--Nijenhuis structures, r-n structures and BN-structures all arise as particular cases, and that the first two correspond to each other on quadratic pre-Leibniz algebras.

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Cite

BibTeXRIS
BibTeX
@misc{benabdelhafidh2026kupershmidtnijenhuis,
  title = {{Kupershmidt--Nijenhuis structures on pre-Leibniz algebras}},
  author = {Sami Benabdelhafidh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40837},
  eprint = {2610.40837},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40837},
}
RIS
TY  - UNPB
TI  - Kupershmidt--Nijenhuis structures on pre-Leibniz algebras
AU  - Sami Benabdelhafidh
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40837
UR  - https://arxiv.org/abs/2610.40837
AB  - In this paper, we first construct the dual of a representation of a pre-Leibniz algebra and study infinitesimal deformations of a pre-Leibniz algebra with a representation, which lead to Nijenhuis pairs, characterizing trivial deformations, and to dual-Nijenhuis pairs. Building on these notions, we introduce Kupershmidt-(dual-)Nijenhuis structures and show that they give rise to hierarchies of pairwise compatible Kupershmidt operators, the converse holding under an invertibility condition. Finally, we show that Rota--Baxter--Nijenhuis structures, r-n structures and BN-structures all arise as particular cases, and that the first two correspond to each other on quadratic pre-Leibniz algebras.
ER  -