Nijenhuis Operators in Pre-Jacobi-Jordan Algebras

• Nijenhuis operators on pre-Jacobi–Jordan algebras are linear maps that intertwine original and deformed multiplications, ensuring cohomological triviality in linear deformations.
• They induce algebraic splits akin to NS and N-dendriform structures while being framed by operadic and graded Lie algebra methods.
• Their study demystifies polynomial identities and operator hierarchies, offering practical insights into deformation theory and integrable systems.

A Nijenhuis operator on a pre-Jacobi–Jordan algebra is a linear endomorphism intertwining the algebra’s nonassociative “pre-Jacobi–Jordan” multiplication with a naturally deformed multiplication, governing the structure and cohomological triviality of linear deformations. The formalism closely connects with broader operator theories in algebra, induces splitting structures reminiscent of NS and N-dendriform algebras, and reflects categorical analogies to Rota–Baxter and pre-Lie settings.

1. Pre-Jacobi–Jordan Algebras: Structure and Identities

A (left) pre-Jacobi–Jordan algebra is an algebra $(A,\cdot)$ such that the product $x \cdot y$ satisfies identities ensuring that the symmetrization $x * y := x \cdot y + y \cdot x$ equips $A$ with the structure of a Jacobi–Jordan algebra (Bremner et al., 2012). The left multiplications $L_\cdot(x): y \mapsto x \cdot y$ yield a representation of the Jacobi–Jordan algebra on $A$, and the defining multilinear identities for $x \cdot y$ (such as those listed in Theorem 4.1 of (Bremner et al., 2012)) guarantee the transfer of the Jacobi–Jordan structure.

Polynomial identities for the pre-Jordan product $x \cdot y = x \succ y + y \prec x$ in dendriform algebras show that while degrees 3 have no nontrivial consequences, degree 4 determines all standard relations up to degree 7, with further exceptional identities in degree 8. These identities underpin the rigidity of the structure and constrain the types of compatible operator actions.

2. Definition and Algebraic Properties of Nijenhuis Operators

Given a pre-Jacobi–Jordan algebra $(A,\cdot)$, a linear map $N: A \to A$ is a Nijenhuis operator if

$$N(x) \cdot N(y) = N(x \cdot_N y), \qquad \forall x, y \in A,$$

where the deformed multiplication is

$$x \cdot_N y = N(x) \cdot y + x \cdot N(y) - N(x \cdot y).$$

This identity, directly mirroring the associative and Lie Nijenhuis operator conditions (see formulas and discussions in (Attan et al., 5 Aug 2025, Baishya et al., 4 May 2025, Bremner et al., 2021)), ensures that the “failure” of $N$ to be multiplicative is precisely captured by the new product.

The operator’s defining property leads to:

• $(A, \cdot_N)$ is itself a pre-Jacobi–Jordan algebra [(Attan et al., 5 Aug 2025), Prop. 6.1].
• $N$ is an algebra morphism from $(A, \cdot_N)$ to $(A, \cdot)$.
• Any polynomial in $N$ yields another Nijenhuis operator (reflecting hierarchical compatibility as in (Baishya et al., 4 May 2025, Yuan, 2022)).

3. Deformation Theory and Cohomological Triviality

Linear deformations of $(A,\cdot)$ are parametrized by

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

where $\omega$ is a bilinear map. For $(A,\cdot_t)$ to remain a pre-Jacobi–Jordan algebra, $\omega$ must satisfy the cohomological identity (be a $2$-cocycle in the relevant cohomology theory developed in (Attan et al., 5 Aug 2025)),

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

If $\omega$ takes this form (i.e., is a coboundary $\delta^1 N$), the deformation is called trivial: the isomorphism $Id_A + tN$ realizes an equivalence between the deformed and original algebra, evidenced by direct algebraic calculation [(Attan et al., 5 Aug 2025), Section 6]. This mechanism matches general deformation theory for associative and Lie algebras (cf. Hochschild cohomology, (Yuan, 2022)).

Thus, Nijenhuis operators are recognized as generators of cohomologically trivial deformations: any first-order deformation generated in this manner does not yield genuinely new algebraic structures.

4. Connections to Rota–Baxter Operators, NS, and N-dendriform Structures

In the associative case, the Nijenhuis operator intertwines with Rota–Baxter operators of weight $-1$ [(Attan et al., 5 Aug 2025), Remark], and the structural splitting into NS and N-dendriform algebras (three operations: $<, >, \cdot$) is obtained via

$$x <_P y = x P(y),\qquad x >_P y = P(x) y,\qquad x \cdot_P y = -P(x y)$$

for Nijenhuis operator $P$ (Guo et al., 2012). These relations satisfy a set of additional quadratic identities—culminating in the N-dendriform algebra, which is structurally richer than the NS algebra.

Such splitting is suggestive in the pre-Jacobi–Jordan context: introducing $<, >, \cdot$-operations via a Nijenhuis-type operator may allow finer classification of algebraic compatibility and facilitate operadic approaches, as the composite identities mirror those found in dendriform, diassociative, and tridendriform settings (Guo et al., 2012, Baishya et al., 4 May 2025).

5. Operadic and Graded Lie Algebra Formulations

The operadic viewpoint uses the Frölicher–Nijenhuis bracket, encoding operators as Maurer–Cartan elements in a differential graded Lie algebra constructed from the nonsymmetric operad with multiplication (Baishya et al., 4 May 2025). For multiplication $\pi$ and $N$ in the operad, the Nijenhuis condition is

$$(\pi \circ_2 N) \circ_1 N = N \circ_1 (\pi \circ_1 N + \pi \circ_2 N - N \circ_1 \pi),$$

with $[N, N]_{FN} = 0$ in the corresponding graded Lie algebra. Power compositions and derived brackets elucidate hierarchies and compatibility among Nijenhuis operators. While this framework is fully established for associative and Loday-type algebras, it provides a template that can be generalized to pre-Jacobi–Jordan algebraic structures.

6. Nijenhuis Operators in Hom and Relative Settings

The Hom-pre-Jacobi–Jordan case integrates a twisting map $\alpha$ and adjusts the above identities: $$N \circ \alpha = \alpha \circ N,\qquad N(x) \cdot N(y) = N( N(x) \cdot y + x \cdot N(y) - N(x \cdot y) )$$ (Attan, 2021). Deformations of the Hom-algebra and symmetrization via the anticommutator link the Nijenhuis operator definition from the pre-algebra to the Jordan context. Matched pairs and representation theory further extend the utility of Nijenhuis operators in constructing new Hom-(pre)-Jacobi–Jordan structures, particularly via semidirect product and lifting procedures.

Relative Rota–Baxter operators in the representation-theoretic setting yield Nijenhuis elements controlling the structure of trivial deformations on semidirect product algebras (Djibril et al., 5 Aug 2025). The relevant zigzag cohomology and compatibility conditions provide a modern algebraic apparatus for analyzing deformations and the interplay of operators in nonassociative contexts.

7. Applications and Further Directions

Nijenhuis operators on pre-Jacobi–Jordan algebras serve as fundamental objects in cohomology, deformation theory, the explicit construction of algebraic splits and compatibility, and the paper of integrable systems and operator hierarchies. Their algebraic roles bridge deformation-theoretic triviality, nontrivial extension to richer quadratic structures, and categorical functorial relationships with NS, N-dendriform, and operadic algebraic frameworks.

The explicit understanding of higher-degree polynomial identities, the characterization of operators via graded Lie algebraic methods, and the utility in computational classification (including symmetry group representation theory (Bremner et al., 2012)) underpin modern explorations of nonassociative algebras and their operator-induced projections. These approaches suggest that Nijenhuis operators will remain central to the ongoing development of pre-Jacobi–Jordan algebra theory, its connections to integrability, and the categorical organization of nonassociative algebraic systems.