Nijenhuis Lie $n$-algebras

We introduce a homotopical framework for Nijenhuis operators on semistrict Lie $n$-algebras ($n$-term $L_\infty$-algebras) for every $n \ge 1$ and, more generally, on arbitrary $L_\infty$-algebras. A Nijenhuis operator is defined as a family $N = (N_q)_{q \ge 1}$ of multilinear maps, with $N_1$ linear, realizing an $L_\infty$-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 $\mathbf{NijLie}_n$; we establish that $\mathbf{NijLie}_m$ embeds as the full subcategory of $\mathbf{NijLie}_n$ on objects concentrated in the bottom $m$ degrees for all $m \le n$. 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 $\mathbf{NijLie}_n$-extensions. Finally, two corrections to published level-three identities are documented, and all component identities up to arity five are verified via computer algebra.