Nijenhuis Operators on Pre-Lie Coalgebras

• The paper presents a dual coalgebraic framework where Nijenhuis operators satisfy a coproduct identity that mirrors the classical Nijenhuis identity on algebras.
• It establishes a correspondence between pre-Lie algebras and coalgebras, enabling explicit constructions using pseudo-Hessian data and quasitriangular structures.
• Low-dimensional examples demonstrate how compatible coproducts and matched pairs yield constrained, well-characterized families of Nijenhuis operators.

Searching arXiv for the cited papers and closely related work on pre-Lie coalgebras, Nijenhuis operators, and pre-Lie bialgebras. Nijenhuis operators on pre-Lie coalgebras are linear endomorphisms of a pre-Lie coalgebra that satisfy a coproduct identity dual to the standard Nijenhuis identity on a pre-Lie algebra. In the framework developed in "Nijenhuis pre-Lie bialgebras, Nijenhuis Lie bialgebras and \sss-equation" (Guo et al., 5 Aug 2025), this notion is not treated as an isolated dualization: it is embedded into pre-Lie bialgebra theory, linked to pseudo-Hessian structures, the $S$-equation and co-$S$-equation, matched pairs, $\mathcal O$-operators, and the passage from pre-Lie to Lie bialgebras. The resulting theory places coalgebra-side Nijenhuis operators within the deformation-theoretic and bialgebraic study of pre-Lie structures.

1. Pre-Lie coalgebras and the coalgebraic Nijenhuis identity

A pre-Lie coalgebra is a pair $(A,\Delta)$ consisting of a vector space $A$ and a linear map

$$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$

written in Sweedler notation. Its defining condition is the pre-Lie coalgebra identity

$$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$

This is the coalgebraic analogue of the pre-Lie identity

$(xy)z-x(yz)=(yx)z-y(xz),$

and expresses that the coassociator is symmetric in the first two tensor slots (Guo et al., 5 Aug 2025).

Within this setting, a Nijenhuis pre-Lie coalgebra is a pair $\big((A,\Delta),S\big)$, where $S:A\to A$ is linear and satisfies

$S$0

This is the coalgebra-side Nijenhuis condition. The corresponding algebra-side identity is

$S$1

The structural interpretation given in the source is that $S$2 controls the failure of $S$3 to be multiplicative on the algebra side, while $S$4 plays the analogous role for the coproduct.

2. Duality with pre-Lie algebras

A central principle is the duality between pre-Lie algebras and pre-Lie coalgebras. If $S$5 is a finite-type pre-Lie coalgebra, then $S$6 carries a pre-Lie algebra structure; conversely, the dual of a pre-Lie algebra structure can be packaged as a coalgebra structure when the pairing is nondegenerate and the data are appropriately finite-dimensional (Guo et al., 5 Aug 2025).

This duality is used in several precise ways. Coalgebraic identities are obtained by dualizing algebraic ones, and vice versa. The coalgebraic Nijenhuis condition is explicitly presented as the direct dual analogue of the algebraic Nijenhuis identity. The same mechanism extends to quasitriangular and dual quasitriangular structures, and to representations and corepresentations: the dual of a representation of a Nijenhuis pre-Lie algebra becomes a representation of the dual algebra under explicit identities. A plausible implication is that the theory is designed so that deformation phenomena on the algebra side admit a systematically mirrored description on the coalgebra side, rather than a merely formal reversal of arrows.

The algebraic background is concretely illustrated in the classification of Nijenhuis operators on $S$7-dimensional pre-Lie algebras over $S$8, where one fixes an isomorphism class $S$9, writes a general linear operator $\mathcal O$0, imposes the Nijenhuis identity on basis pairs, and solves the resulting polynomial equations (Basdouri et al., 28 Apr 2025). That classification is algebraic rather than coalgebraic, but it provides a finite-dimensional model for the kind of explicit rigidity that the dual theory later exploits.

3. Pre-Lie bialgebras, quasitriangularity, and the $\mathcal O$1-equation

The coalgebraic notion is developed inside the theory of pre-Lie bialgebras. A pre-Lie bialgebra $\mathcal O$2 is simultaneously a pre-Lie algebra and a pre-Lie coalgebra, with compatibility conditions

$\mathcal O$3

and

$\mathcal O$4

These are the pre-Lie analogues of Lie bialgebra compatibility relations (Guo et al., 5 Aug 2025).

For a pre-Lie algebra $\mathcal O$5 and $\mathcal O$6, the coproduct

$\mathcal O$7

produces a quasitriangular pre-Lie bialgebra when $\mathcal O$8 is a symmetric solution of the $\mathcal O$9-equation

$(A,\Delta)$0

The paper also gives the equivalent operator characterizations

$(A,\Delta)$1

and, if $(A,\Delta)$2 is symmetric,

$(A,\Delta)$3

These constructions are foundational because the later coalgebraic Nijenhuis operators are obtained from pre-Lie bialgebra data. The significance is not only that bialgebras supply examples, but that they organize the compatibility conditions under which an operator on the coalgebra side is natural.

4. Dual quasitriangularity, the co-$(A,\Delta)$4-equation, and pseudo-Hessian data

The dual side begins with a symmetric bilinear form $(A,\Delta)$5 on a pre-Lie coalgebra $(A,\Delta)$6. It is a solution of the co-$(A,\Delta)$7-equation if

$(A,\Delta)$8

for all $(A,\Delta)$9. From such a symmetric solution the paper defines

$A$0

Then $A$1 is a pre-Lie bialgebra, called dual quasitriangular (Guo et al., 5 Aug 2025).

The co-$A$2-equation is equivalent to

$A$3

and, when $A$4 is symmetric, dual quasitriangularity is equivalent to

$A$5

These identities are presented as exact dual mirrors of the quasitriangular operator characterizations.

The same section places pseudo-Hessian structures on both sides. A pseudo-Hessian pre-Lie algebra is a pre-Lie algebra $A$6 equipped with a symmetric bilinear form satisfying

$A$7

Its coalgebra analogue is a pseudo-Hessian pre-Lie coalgebra, defined by a symmetric tensor

$A$8

satisfying

$A$9

The paper uses this pseudo-Hessian coalgebraic input in its principal construction of Nijenhuis operators on pre-Lie coalgebras.

5. Construction methods for Nijenhuis operators on pre-Lie coalgebras

The paper emphasizes two main construction methods for Nijenhuis operators on pre-Lie coalgebras (Guo et al., 5 Aug 2025).

The first starts from linearly compatible pre-Lie coalgebras. Two coproducts $$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$0 and $$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$1 are linearly compatible if every linear combination

$$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$2

is again a pre-Lie coalgebra for all $$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$3. This is equivalent to the mixed identity

$$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$4

$$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$5

If $$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$6 and $$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$7 are linearly compatible and there is a corepresentation homomorphism of the form

$$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$8

then the compatibility equations imply that $$\Delta:A\to A\otimes A,\qquad \Delta(x)=x_{(1)}\otimes x_{(2)},$$9 is a Nijenhuis operator on $$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$0. The theorem gives, in particular,

$$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$1

and

$$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$2

This is explicitly described as the coalgebra analogue of the principle that compatible algebra structures produce Nijenhuis operators.

The second method is the dual of the algebraic construction from pseudo-Hessian and quasitriangular data. If $$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$3 is a pseudo-Hessian pre-Lie coalgebra with symmetric tensor $$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$4, define

$$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$5

If, in addition, $$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$6 is a dual quasitriangular pre-Lie bialgebra and $$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$7 is a quasitriangular pre-Lie bialgebra, then $$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$8 is a Nijenhuis pre-Lie coalgebra. The source identifies this theorem as the direct dual counterpart of the algebraic construction

$$x_{(1)(1)}\,x_{(1)(2)}\,x_{(2)}-x_{(1)}\,x_{(2)(1)}\,x_{(2)(2)} = x_{(1)(2)}\,x_{(1)(1)}\,x_{(2)}-x_{(2)(1)}\,x_{(1)}\,x_{(2)(2)}. \tag{CL}$$9

6. Nijenhuis pre-Lie bialgebras, $(xy)z-x(yz)=(yx)z-y(xz),$0-operators, and passage to Lie bialgebras

A Nijenhuis pre-Lie bialgebra is defined as a quintuple

$(xy)z-x(yz)=(yx)z-y(xz),$1

such that $(xy)z-x(yz)=(yx)z-y(xz),$2 is a Nijenhuis pre-Lie algebra, $(xy)z-x(yz)=(yx)z-y(xz),$3 is a Nijenhuis pre-Lie coalgebra, $(xy)z-x(yz)=(yx)z-y(xz),$4 is a pre-Lie bialgebra, $(xy)z-x(yz)=(yx)z-y(xz),$5 is $(xy)z-x(yz)=(yx)z-y(xz),$6-admissible, and the dual representation $(xy)z-x(yz)=(yx)z-y(xz),$7 is $(xy)z-x(yz)=(yx)z-y(xz),$8-admissible (Guo et al., 5 Aug 2025). The admissibility conditions include

$(xy)z-x(yz)=(yx)z-y(xz),$9

and

$\big((A,\Delta),S\big)$0

A key theorem states that a Nijenhuis pre-Lie bialgebra is equivalent to a matched pair of Nijenhuis pre-Lie algebras,

$\big((A,\Delta),S\big)$1

together with the usual matched-pair conditions and the Nijenhuis compatibilities.

The same framework enriches the $\big((A,\Delta),S\big)$2-equation by a Nijenhuis operator. If $\big((A,\Delta),S\big)$3 is a symmetric solution of the $\big((A,\Delta),S\big)$4-equation and also satisfies

$\big((A,\Delta),S\big)$5

then $\big((A,\Delta),S\big)$6 solves the $\big((A,\Delta),S\big)$7-Nijenhuis $\big((A,\Delta),S\big)$8-equation. For symmetric $\big((A,\Delta),S\big)$9, this is equivalent to the conditions that $S:A\to A$0 is a weak $S:A\to A$1-operator associated to $S:A\to A$2 and $S:A\to A$3, and that

$S:A\to A$4

If the pre-Lie algebra is $S:A\to A$5-admissible, the weak $S:A\to A$6-operator becomes an actual $S:A\to A$7-operator associated to the Nijenhuis representation. The paper then proves that $S:A\to A$8-operators produce symmetric solutions on semidirect products $S:A\to A$9, yielding Nijenhuis pre-Lie bialgebras.

The final transition is to Lie theory by antisymmetrization. Given a pre-Lie bialgebra $S$00, define

$S$01

The pre-Lie bialgebra is called balanced if

$S$02

The theorem stated in the source is that $S$03 is a Lie bialgebra if and only if the original pre-Lie bialgebra is balanced. Moreover, if $S$04 is a Nijenhuis pre-Lie bialgebra and $S$05 is balanced, then

$S$06

is a Nijenhuis Lie bialgebra. This passage is significant because it converts the pre-Lie and coalgebraic Nijenhuis data into Lie-bialgebraic examples.

7. Low-dimensional examples and conceptual significance

The paper gives several low-dimensional examples on a $S$07-dimensional vector space $S$08. These examples exhibit explicit symmetric solutions of the co-$S$09-equation, explicit pseudo-Hessian pre-Lie algebra structures, explicit Nijenhuis operators obtained from the algebraic construction and its coalgebraic dual, and examples of Nijenhuis pre-Lie bialgebras and Nijenhuis Lie bialgebras (Guo et al., 5 Aug 2025). In the simplest nondegenerate case, the operator constructed from a symmetric solution $S$10 becomes the identity $S$11, showing that the general construction recovers the obvious Nijenhuis operator in the invertible case. The final examples are stated to be balanced and therefore yield explicit Nijenhuis Lie bialgebras.

From the algebraic side, the classification of Nijenhuis operators on $S$12-dimensional complex pre-Lie algebras shows that the operator families are typically sparse and highly constrained: once one fixes one of the eight isomorphism classes $S$13, the Nijenhuis identity reduces to a small polynomial system whose solutions form a short list of matrix families (Basdouri et al., 28 Apr 2025). This suggests an analogous expectation for coalgebra-side constructions in low dimension: explicit formulas are often possible, but only under strong structural constraints.

A common misunderstanding would be to regard Nijenhuis operators on pre-Lie coalgebras as merely the formal dual of an already known algebraic identity. The framework of (Guo et al., 5 Aug 2025) shows a stronger statement: the coalgebraic notion is integrated with linearly compatible coproducts, pseudo-Hessian tensors, quasitriangular and dual quasitriangular pre-Lie bialgebras, the co-$S$14-equation, $S$15-operators, matched pairs, and the construction of Nijenhuis Lie bialgebras. In that sense, the coalgebraic theory is a structural component of a broader pre-Lie deformation and bialgebra program rather than an isolated dual definition.