Nijenhuis Operators on Pre-Jacobi–Jordan Algebras

• Nijenhuis operators on pre-Jacobi–Jordan algebras are linear endomorphisms that enforce controlled deformations through a defined quadratic identity.
• They induce twisted products that preserve algebraic identities and yield trivial linear deformations via explicit automorphisms.
• These operators underpin cohomological, operadic, and categorical constructions, facilitating systematic studies in nonassociative algebra deformations.

A Nijenhuis operator on a pre-Jacobi–Jordan algebra is a linear endomorphism that produces a controlled deformation of the algebraic structure through a specific operator identity, generalizing classical notions from the associative, Lie, and pre-Lie settings. Recent work has clarified their precise definition, structural role, and consequences for cohomology and deformation theory in both standard and Hom-variant settings. Such operators are a central tool for understanding trivial deformations, operadic splittings, and category-theoretic functorial constructions in pre-Jacobi–Jordan algebra theory.

1. Definition of Nijenhuis Operator on Pre-Jacobi–Jordan Algebras

Let $(A, \cdot)$ be a left pre-Jacobi–Jordan algebra over a field $k$. A linear operator $N: A \rightarrow A$ is called a Nijenhuis operator if

$$N(x) \cdot N(y) = N\big( N(x) \cdot y + x \cdot N(y) - N(x \cdot y) \big)$$

for all $x, y \in A$ (Attan et al., 5 Aug 2025). This compatibility condition expresses that the product of images under $N$ is again the image under $N$ of a sum constructed from the operator and the original product—making $N$ an intertwiner of the algebra’s binary operation and its own induced twist.

The operator $N$ gives rise to an $\boldsymbol{N}$-twisted product: $$x \cdot_N y := N(x) \cdot y + x \cdot N(y) - N(x \cdot y),$$ which induces a new pre-Jacobi–Jordan algebra structure $(A, \cdot_N)$.

2. Structure, Morphisms, and Twisted Products

If $N$ is a Nijenhuis operator on $(A, \cdot)$, then $(A, \cdot_N)$ is again a left pre-Jacobi–Jordan algebra. Furthermore,

$N(x \cdot_N y) = N(x) \cdot N(y),$

meaning $N$ is a morphism from $(A, \cdot_N)$ to $(A, \cdot)$. This morphism property guarantees that the deformed product $\cdot_N$ is compatible with the original product $\cdot$ through $N$, and algebraic identities (including Jacobi-type or Jordan-type symmetries) are preserved when viewed through $N$.

3. Cohomological Role and Triviality of Deformations

In deformation theory, a linear deformation is expressed as

$x \cdot_t y = x \cdot y + t\,\omega(x, y)$

with $t$ a formal parameter and $\omega$ a 2-cocycle in the cohomology theory of pre-Jacobi–Jordan algebras (Attan et al., 5 Aug 2025). If $N$ is a Nijenhuis operator, then

$$\omega(x, y) := x \cdot_N y = N(x) \cdot y + x \cdot N(y) - N(x \cdot y)$$

is a 2-cocycle, and the corresponding deformation

$x \cdot_t y = x \cdot y + t\,\omega(x, y)$

is trivial. Explicitly, the linear map $\Phi_t = \mathrm{Id}_A + t N$ is an isomorphism between $(A, \cdot_t)$ and $(A, \cdot)$: $\Phi_t(x \cdot_t y) = \Phi_t(x) \cdot \Phi_t(y).$ This result parallels the classical observation in associative and Lie settings: a Nijenhuis operator generates only coordinate changes, not genuinely new algebraic structures.

4. Operadic and Category-Theoretic Perspectives

In associative and related settings (see (Guo et al., 2012)), Nijenhuis operators facilitate an adjoint functor construction: free Nijenhuis algebras are built from associative or NS algebras by explicit bracketed word combinatorics and ideal quotienting. These functorial constructions extend naturally to pre-Jacobi–Jordan algebras in form, allowing realization of universal enveloping Nijenhuis algebras in potential categorical frameworks. The principle is that an operator splitting via a Nijenhuis operator yields algebraic structures compatible with existing identities but possibly governed by refined quadratic relations, as for N-dendriform and NS algebras (Guo et al., 2012).

5. Relations to Pre-Jordan, Dendriform, and Related Structures

In dendriform and pre-Jordan settings (Bremner et al., 2012), operator splittings using Nijenhuis-type conditions induce rich combinatorial identities. For pre-Jacobi–Jordan algebras, the analogous twist by $N$ maintains symmetry properties and could produce splitting phenomena: $$x \lesssim y = x N(y),\quad x \gtrsim y = N(x) y,\quad x \odot y = -N(x \cdot y)$$ (following the construction motifs in (Guo et al., 2012)). Such splits may allow systematic identification of new algebraic identities, possibly subject to additional constraints in higher tensor degrees, as seen for degree-8 identities in the pre-Jordan case.

6. Hom-Type and Relative Setting

The Hom-pre-Jacobi–Jordan case generalizes the operator identity to include a twisting map $\alpha$ (Attan, 2021). For a Hom-Jacobi–Jordan algebra $(A, *, \alpha)$, the Nijenhuis condition becomes

$$N(x) * N(y) = N\left( N(x) * y + x * N(y) - N(x * y) \right)$$

with the requirement $N\alpha = \alpha N$.

In the relative Rota-Baxter framework (Djibril et al., 5 Aug 2025), Nijenhuis elements characterize trivial linear deformations of relative Rota-Baxter operators, confirming their role in “gauge equivalence” classes of deformations and in the full cohomological control of (pre-)Jacobi–Jordan algebra operator deformations.

7. Examples, Classification, and Computational Aspects

For low-dimensional pre-Lie algebras, explicit matrix classification of Nijenhuis operators is achieved by direct computation (Basdouri et al., 28 Apr 2025). For 2-dimensional cases, the only nontrivial Nijenhuis operator up to equivalence is upper-triangular with a single off-diagonal entry; analogous matrix computations could be carried out for pre-Jacobi–Jordan algebras with known multiplication tables.

In the context of algebraic operads, the Maurer–Cartan equation for the Frölicher–Nijenhuis bracket provides a universal characterization of Nijenhuis operators. For a nonsymmetric operad with multiplication $\pi$,

$$[N, N]_{FN} = 0 \quad \iff \quad N(x) \cdot N(y) = N(N(x) \cdot y + x \cdot N(y) - N(x \cdot y))$$

(Baishya et al., 4 May 2025). This generalizes to all Loday-type algebras, including pre-Jacobi–Jordan algebras.

Conclusion

Nijenhuis operators on pre-Jacobi–Jordan algebras intertwine the original and twisted products by a specific quadratic identity, generalizing classical integrability and deformation-theoretic notions. They induce new algebraic structures that are isomorphic to the original via explicit automorphisms, ensuring the corresponding linear deformations are trivial. Their universal characterization via operadic and cohomological methods, compatibility with categorical constructions, and combinatorial splittings connect them with broader developments in nonassociative algebra, deformation theory, and algebraic operad theory. This conceptual and technical framework enables systematic paper of deformations, representations, and operator-induced hierarchies in pre-Jacobi–Jordan and related algebraic systems.