Nijenhuis Operators on Pre-Lie Algebras

• The topic defines a Nijenhuis operator on a pre-Lie algebra by specifying that its torsion vanishes, ensuring a trivial deformation of the algebraic structure.
• It details the construction of a Nijenhuis hierarchy, linking the operator with pseudo-Hessian forms, S-equations, and O-operator methodologies.
• It explores the geometric interpretation at scalar-type singular points, where the tangent space inherits a pre-Lie structure, thereby connecting pre-Lie and Lie theories.

A Nijenhuis operator on a pre-Lie algebra is a linear endomorphism whose torsion vanishes in a way compatible with the nonassociative left-symmetric product. In the pre-Lie setting, such operators are tied simultaneously to deformation theory, pseudo-Hessian geometry, pre-Lie bialgebras, $S$-equations, $\mathcal O$-operators, and the passage from pre-Lie structures to Lie-theoretic ones (Guo et al., 5 Aug 2025, Wang et al., 2017). The subject also has a geometric counterpart: at scalar-type singular points of a Nijenhuis tensor field, the tangent space acquires a natural left-symmetric algebra structure, and the local linearization problem for Nijenhuis operators can be reformulated in terms of the resulting pre-Lie algebra (Konyaev, 2019).

1. Basic definition and algebraic consequences

A left pre-Lie algebra is a pair $(A,\cdot)$ where $A$ is a vector space and $\cdot:A\otimes A\to A$ is bilinear, satisfying

$$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$

The same structures are also referred to in the literature as left-symmetric algebras, pre-Lie algebras, or Vinberg-Koszul algebras (Guo et al., 5 Aug 2025, Konyaev, 2019).

A linear endomorphism $N:A\to A$ is called a Nijenhuis operator if

$$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$

Equivalently, one introduces the Nijenhuis torsion

$$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$

and the Nijenhuis condition is exactly $T_N(x,y)=0$ for all $\mathcal O$0 (Guo et al., 5 Aug 2025).

If $\mathcal O$1 is a Nijenhuis operator on $\mathcal O$2, then

$\mathcal O$3

defines another pre-Lie product on $\mathcal O$4. More generally,

$\mathcal O$5

gives a sequence of operations “all of which mutually commute (in the sense that mixed associators vanish).” The paper describes this as the standard “Nijenhuis hierarchy” of commuting flows in integrable systems theory (Guo et al., 5 Aug 2025).

The induced Lie bracket

$\mathcal O$6

plays a persistent role. It is the subadjacent Lie algebra of the pre-Lie algebra, and many constructions for Nijenhuis operators on pre-Lie algebras pass through this Lie-admissible bracket (Gao et al., 2023).

2. Deformation-theoretic formulation

A central structural result is that pre-Lie products can be encoded by a graded Lie algebra. For a fixed finite-dimensional vector space $\mathcal O$7, one sets

$\mathcal O$8

and equips $\mathcal O$9 with a graded Lie bracket $(A,\cdot)$0. A bilinear map $(A,\cdot)$1 defines a pre-Lie product $(A,\cdot)$2 if and only if

$(A,\cdot)$3

that is, $(A,\cdot)$4 is a Maurer-Cartan element in $(A,\cdot)$5 (Wang et al., 2017).

Within this framework, Nijenhuis operators are precisely the operators generating trivial deformations. For a fixed pre-Lie algebra $(A,\cdot)$6, the deformation determined by $(A,\cdot)$7 is

$(A,\cdot)$8

If $(A,\cdot)$9 is Nijenhuis, then $A$0 is a pre-Lie algebra for all $A$1, and the family of linear maps

$A$2

carries $A$3 back to $A$4. In this sense, every Nijenhuis operator generates a trivial infinitesimal-deformation class in $A$5 (Wang et al., 2017).

This formulation clarifies why Nijenhuis operators are not merely auxiliary endomorphisms. They encode a mechanism by which the pre-Lie product changes while remaining within the same deformation class. A plausible implication is that the Nijenhuis condition should be viewed less as an isolated quadratic identity and more as a compatibility condition between a pre-Lie structure and its trivialized deformation complex.

3. Construction from pseudo-Hessian pre-Lie algebras and the $A$6-equation

A triple $A$7 is called a pseudo-Hessian pre-Lie algebra if $A$8 is a pre-Lie algebra, $A$9 is a nondegenerate symmetric bilinear form, and

$\cdot:A\otimes A\to A$0

Equivalently, $\cdot:A\otimes A\to A$1 is a $\cdot:A\otimes A\to A$2-cocycle in the trivial $\cdot:A\otimes A\to A$3-bimodule $\cdot:A\otimes A\to A$4 (Guo et al., 5 Aug 2025).

The paper "Nijenhuis pre-Lie bialgebras, Nijenhuis Lie bialgebras and $\cdot:A\otimes A\to A$5-equation" gives a direct construction of Nijenhuis operators from this data. If

$\cdot:A\otimes A\to A$6

is symmetric and satisfies the pre-Lie $\cdot:A\otimes A\to A$7-equation, then

$\cdot:A\otimes A\to A$8

defines a Nijenhuis operator on $\cdot:A\otimes A\to A$9. The associated coproduct is

$$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$0

and $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$1 is a quasitriangular pre-Lie bialgebra. The theorem recorded in the paper states that if $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$2 is pseudo-Hessian and $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$3 is a symmetric solution of the $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$4-equation, then the above $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$5 satisfies the Nijenhuis identity (Guo et al., 5 Aug 2025).

The same circle of ideas appears in the earlier operator-theoretic treatment. For the dual representation of the regular representation of a pre-Lie algebra, an $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$6-operator is precisely an $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$7-matrix $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$8 satisfying the $$(x\cdot y)\cdot z - x\cdot(y\cdot z) = (y\cdot x)\cdot z - y\cdot(x\cdot z), \qquad \forall x,y,z\in A.$$9-equation, and the inverse $N:A\to A$0 is a pseudo-Hessian form $N:A\to A$1. In this setting, compatible invertible $N:A\to A$2-matrices $N:A\to A$3 yield a pseudo-Hessian-Nijenhuis structure with

$N:A\to A$4

and the forms

$N:A\to A$5

are again pseudo-Hessian forms (Wang et al., 2017).

The explicit two-dimensional example in (Guo et al., 5 Aug 2025) makes the construction concrete. Let $N:A\to A$6 with

$N:A\to A$7

Define

$N:A\to A$8

and take

$N:A\to A$9

Then $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$0 is pseudo-Hessian, $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$1 is symmetric and satisfies the $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$2-equation, and the induced operator is

$$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$3

A direct verification shows that $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$4 is Nijenhuis (Guo et al., 5 Aug 2025).

4. Relations with $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$5-operators, Rota-Baxter operators, L-dendriform structures, and bialgebras

There are close relationships between $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$6-operators, Rota-Baxter operators, and Nijenhuis operators on a pre-Lie algebra (Wang et al., 2017). If $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$7 is a representation of a pre-Lie algebra $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$8, an $$N(x)\cdot N(y) = N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr), \qquad \forall x,y\in A.$$9-operator $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$0 satisfies

$$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$1

A Rota-Baxter operator of weight $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$2 on $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$3 is a linear map $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$4 such that

$$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$5

The paper proves, among other facts, that every Rota-Baxter operator of weight $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$6 is an $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$7-operator for the regular representation; if $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$8, then $$T_N(x,y) = N(x)\cdot N(y) - N\bigl(N(x)\cdot y + x\cdot N(y) - N(x\cdot y)\bigr),$$9 is Nijenhuis if and only if $T_N(x,y)=0$0 is a Rota-Baxter operator of weight $T_N(x,y)=0$1; and if $T_N(x,y)=0$2, then $T_N(x,y)=0$3 is Nijenhuis if and only if $T_N(x,y)=0$4 is a Rota-Baxter operator of weight $T_N(x,y)=0$5 (Wang et al., 2017).

The same work also shows that a Nijenhuis operator “connects” two $T_N(x,y)=0$6-operators on a pre-Lie algebra whose any linear combination is still an $T_N(x,y)=0$7-operator in certain sense. If $T_N(x,y)=0$8 is an $T_N(x,y)=0$9-operator and $\mathcal O$00 is Nijenhuis with

$\mathcal O$01

then $\mathcal O$02 is again an $\mathcal O$03-operator; if $\mathcal O$04 is invertible, then $\mathcal O$05 and $\mathcal O$06 are compatible $\mathcal O$07-operators, and compatible L-dendriform algebras appear naturally as the induced algebraic structures (Wang et al., 2017).

The bialgebraic side is developed further in (Guo et al., 5 Aug 2025). That paper introduces the notion of Nijenhuis operators on pre-Lie coalgebras and gives two constructions, “one from a linearly compatible pre-Lie coalgebra structure, and one from pre-Lie bialgebras.” It then obtains a bialgebraic structure on Nijenhuis pre-Lie algebras by using dual representations and studies their relations with $\mathcal O$08-equations and $\mathcal O$09-operators. Its final Lie-theoretic conclusion is that “a Nijenhuis balanced pre-Lie bialgebra produces a Nijenhuis Lie bialgebra” (Guo et al., 5 Aug 2025).

These results place Nijenhuis operators at an intersection point. They are simultaneously deformation operators, operator-theoretic mediators between different $\mathcal O$10-operators, and algebraic devices for transferring structure from the pre-Lie level to the Lie bialgebra level.

5. Geometric interpretation and the linearization problem

In Nijenhuis geometry, one studies a $\mathcal O$11-tensor field

$\mathcal O$12

whose Nijenhuis torsion vanishes. A point $\mathcal O$13 is of scalar type if

$\mathcal O$14

for some $\mathcal O$15. At such a point, the tangent space carries a natural left-symmetric algebra structure: $\mathcal O$16 Equivalently,

$\mathcal O$17

In coordinates centered at $\mathcal O$18 with $\mathcal O$19, the structure constants are

$\mathcal O$20

Thus the tangent space at a scalar-type point possesses a natural structure of a left-symmetric algebra (Konyaev, 2019).

This observation leads to the linearization problem for Nijenhuis operators. If $\mathcal O$21 vanishes at $\mathcal O$22 and $\mathcal O$23 is its first-order term, one asks when there is a local change of coordinates making $\mathcal O$24 exactly equal to $\mathcal O$25. Since $\mathcal O$26 is the right-adjoint of the isotropy left-symmetric algebra, the problem becomes one of deciding when the isotropy pre-Lie algebra completely determines the local form of the Nijenhuis operator (Konyaev, 2019).

Konyaev defines a finite-dimensional left-symmetric algebra $\mathcal O$27 to be nondegenerate if any Nijenhuis operator whose isotropy algebra at a scalar-type point is isomorphic to $\mathcal O$28 is linearizable. In dimension $\mathcal O$29, the paper gives a complete classification of two-dimensional real left-symmetric algebras and shows that the smooth and analytic categories differ. In the smooth category, the degenerate algebras are exactly

$\mathcal O$30

where

$\mathcal O$31

In the analytic category, the same list is degenerate with

$\mathcal O$32

and the remaining two-dimensional LSAs are nondegenerate except the countable set of “Yoccoz-type” parameters $\mathcal O$33, which remain open (Konyaev, 2019).

A common simplification is to treat smooth and analytic linearization as parallel. The two-dimensional classification shows that “these two cases, analytic and smooth, differ” (Konyaev, 2019). This suggests that the analytic behavior of Nijenhuis operators is sensitive not only to the pre-Lie isotropy algebra but also to small-divisor phenomena encoded in the eigen-ratio arithmetic.

6. Low-dimensional classifications, CYBE constructions, and equivariant extensions

Low-dimensional classification results make the abstract theory explicit. Gao, Kang, Lü, and Yu classify all Nijenhuis operators on $\mathcal O$34-dimensional complex pre-Lie algebras and on $\mathcal O$35-dimensional complex associative algebras, and they use the obtained classification to provide solutions of the classical Yang-Baxter equation of the semidirect product sub-adjacent Lie algebras (Gao et al., 2023). In their $\mathcal O$36-dimensional commutative case $\mathcal O$37, the zero algebra, any linear map is Nijenhuis. In the noncommutative case $\mathcal O$38, one finds

$\mathcal O$39

When $\mathcal O$40, the corresponding Rota-Baxter operator of weight $\mathcal O$41 on the sub-adjacent Lie algebra gives a skew solution

$\mathcal O$42

of the classical Yang-Baxter equation in the semidirect-product double (Gao et al., 2023).

Basdouri, Mosbahi, and Zahari study Rota-Baxter, Reynolds, Nijenhuis, and Averaging operators on $\mathcal O$43-dimensional pre-Lie algebras over $\mathcal O$44, using the classification of $\mathcal O$45-dimensional pre-Lie algebras and computational tools like Mathematica or Maple (Basdouri et al., 28 Apr 2025). In their list of eight isomorphism classes, the Nijenhuis operators on $\mathcal O$46 are precisely the arbitrary diagonal operators

$\mathcal O$47

while on $\mathcal O$48 they satisfy

$\mathcal O$49

together with

$\mathcal O$50

These classifications show how the Nijenhuis condition reduces, in low dimension, to concrete polynomial constraints on matrix entries (Basdouri et al., 28 Apr 2025).

A recent extension places Nijenhuis operators on pre-Lie algebras into an equivariant Lie-theoretic framework. In this setting, a strong pre-ENL algebra is a triple $\mathcal O$51 such that $\mathcal O$52 is strongly equivariant: $\mathcal O$53 Strong equivariance implies weak equivariance on the induced Lie bracket,

$\mathcal O$54

but the converse does not hold in general. If $\mathcal O$55 is a pre-ENL algebra, then $\mathcal O$56 is an ENL algebra, the left multiplication $\mathcal O$57 defines an EN-representation, and the identity $\mathcal O$58 is an EN-relative Rota-Baxter operator of weight $\mathcal O$59 (Hou et al., 27 Jan 2026).

The same equivariant framework yields a canonical Yang-Baxter object. On the semidirect product ENL algebra

$\mathcal O$60

the tensor

$\mathcal O$61

is a skew-symmetric solution of the classical Yang-Baxter equation compatible with $\mathcal O$62, hence an EN $\mathcal O$63-matrix (Hou et al., 27 Jan 2026). This suggests that the operator theory of Nijenhuis pre-Lie algebras continues to expand toward matched pairs, doubles, and operator-equipped bialgebra structures rather than remaining confined to deformation theory alone.