Nijenhuis Lie n-algebras
Abstract
We introduce a homotopical framework for Nijenhuis operators on semistrict Lie n-algebras (n-term L∞-algebras) for every n ≥ 1 and, more generally, on arbitrary L∞-algebras. A Nijenhuis operator is defined as a family N = (Nq)q ≥ 1 of multilinear maps, with N1 linear, realizing an L∞-morphism from an N-twisted copy of the structure to the original one. We prove that this condition is equivalent to the vanishing, at every arity m, of the m-th Nijenhuis torsion up to a correction term ΛNm formed by higher-component substitutions, and we provide a closed combinatorial formula for ΛNm as a sum over set partitions. Along the way, we show that the arity-k Nijenhuis torsion factors canonically as a composite of k commuting elementary slot deformations. The resulting objects form a category NijLien; we establish that NijLiem embeds as the full subcategory of NijLien on objects concentrated in the bottom m degrees for all m ≤ n. These levels assemble into a tower whose first three stages are classical Nijenhuis Lie algebras, the Nijenhuis Lie $2$-algebras of Das, and the newly introduced Nijenhuis Lie $3$-algebras. We characterize the strict and skeletal objects at every level, introduce n-term representations up to homotopy of a Nijenhuis Lie algebra, and prove that the semidirect product construction establishes an equivalence between the category of such representations and split abelian NijLien-extensions. Finally, two corrections to published level-three identities are documented, and all component identities up to arity five are verified via computer algebra.
Related
- Kupershmidt--Nijenhuis structures on pre-Leibniz algebras
- Classification of Nijenhuis operators on three-dimensional multiplicative simple Bihom-Lie algebras
- A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras
- From quadratic integrals to Nijenhuis operators: two-dimensional dictionary
- Nijenhuis torsion and Frölicher-Nijenhuis brackets of recursion operators via their full-fledged forms
BibTeX
@misc{basdouri2026nijenhuis,
title = {{Nijenhuis Lie $n$-algebras}},
author = {Imed Basdouri and Tarek Ben Massoud and Ahmed Zahari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37698},
eprint = {2610.37698},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37698},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nijenhuis Lie n-algebras
AU - Imed Basdouri
AU - Tarek Ben Massoud
AU - Ahmed Zahari
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.37698
UR - https://arxiv.org/abs/2610.37698
AB - We introduce a homotopical framework for Nijenhuis operators on semistrict Lie n-algebras (n-term L_∞-algebras) for every n ≥ 1 and, more generally, on arbitrary L_∞-algebras. A Nijenhuis operator is defined as a family N = (N_q)_q ≥ 1 of multilinear maps, with N_1 linear, realizing an L_∞-morphism from an N-twisted copy of the structure to the original one. We prove that this condition is equivalent to the vanishing, at every arity m, of the m-th Nijenhuis torsion up to a correction term Λ^N_m formed by higher-component substitutions, and we provide a closed combinatorial formula for Λ^N_m as a sum over set partitions. Along the way, we show that the arity-k Nijenhuis torsion factors canonically as a composite of k commuting elementary slot deformations. The resulting objects form a category NijLie_n; we establish that NijLie_m embeds as the full subcategory of NijLie_n on objects concentrated in the bottom m degrees for all m ≤ n. These levels assemble into a tower whose first three stages are classical Nijenhuis Lie algebras, the Nijenhuis Lie $2$-algebras of Das, and the newly introduced Nijenhuis Lie $3$-algebras. We characterize the strict and skeletal objects at every level, introduce n-term representations up to homotopy of a Nijenhuis Lie algebra, and prove that the semidirect product construction establishes an equivalence between the category of such representations and split abelian NijLie_n-extensions. Finally, two corrections to published level-three identities are documented, and all component identities up to arity five are verified via computer algebra.
ER -