NS-3-Leibniz Algebras: An Operator Splitting Approach

• NS-3-Leibniz algebras are ternary Leibniz structures that decompose a single 3-Leibniz bracket into four distinct, compatible trilinear operations.
• They are derived from twisted Rota-Baxter operators on 3-Leibniz algebras, paralleling the binary NS-Leibniz splitting paradigm.
• Concrete operator examples like Nijenhuis and Reynolds operators illustrate the practical use of NS-3-Leibniz algebras in deformation theory and other algebraic frameworks.

NS-3-Leibniz algebras are ternary Leibniz-type structures equipped with four trilinear operations that refine a single 3-Leibniz bracket into a compatible “split” system. In the formulation introduced in "Twisted Rota-Baxter operators on 3-Leibniz algebras and NS-3-Leibniz algebras" (Teng, 30 Jul 2025), an NS-3-Leibniz algebra is the underlying algebraic structure of twisted Rota-Baxter operators on 3-Leibniz algebras. Its basic role is therefore analogous to the role played by NS-Leibniz algebras in the binary setting, where a Leibniz bracket is decomposed into several compatible operations arising from twisted relative Rota-Baxter operators (Das et al., 2021).

1. Position within ternary Leibniz theory

The ambient structure for NS-3-Leibniz theory is a 3-Leibniz algebra, namely a vector space $\mathfrak g$ with a trilinear bracket

$$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$

satisfying the 3-Leibniz identity

$$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$

This is the only defining identity required in the source that introduces NS-3-Leibniz algebras (Teng, 30 Jul 2025).

The broader ternary landscape already contains several related notions. A left Leibniz 3-algebra, used in finiteness theorems of Schur type, is defined by the same formal pattern of derivation in the third slot (Minaiev et al., 2024). A 3-Leibniz-Lie algebra, introduced as the algebraic structure underlying a nonabelian embedding tensor on a 3-Lie algebra, consists of a 3-Lie bracket together with an additional ternary product satisfying mixed compatibility identities (Teng et al., 2023). A 3-tri-Leibniz algebra, induced by embedding tensors on 3-Leibniz algebras, splits ternary structure into three trilinear operations $[\cdot,\cdot,\cdot]_{\vdash}$, $[\cdot,\cdot,\cdot]_{\dashv}$, and $[\cdot,\cdot,\cdot]_{\perp}$ (Teng et al., 6 Feb 2025). NS-3-Leibniz algebras belong to this same family of split ternary Leibniz-type systems, but with four operations rather than three (Teng, 30 Jul 2025).

A plausible implication is that NS-3-Leibniz theory occupies the ternary analogue of the dendriform/NS-style splitting paradigm: one begins with a single 3-Leibniz bracket and resolves it into multiple operations constrained so that their sum remains 3-Leibniz. This is explicit in the binary precursor, where NS-Leibniz algebras split a Leibniz algebra into three products $\triangleright$, $\triangleleft$, and $\diamond$ (Das et al., 2021).

2. Twisted Rota-Baxter origin

The direct source of NS-3-Leibniz algebras is the theory of twisted Rota-Baxter operators on 3-Leibniz algebras (Teng, 30 Jul 2025). Let $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a 3-Leibniz algebra, let $$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$0 be a representation, and let

$$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$1

be a $$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$2-cocycle in the 3-Leibniz cohomology of $$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$3 with coefficients in $$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$4. A linear map

$$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$5

is called a $$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$6-twisted Rota-Baxter operator if

$$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$7

for all $$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$8 (Teng, 30 Jul 2025).

This operator has a graph characterization. Writing

$$[\cdot,\cdot,\cdot]_{\mathfrak g}:\mathfrak g^{\otimes 3}\to \mathfrak g$$9

the paper proves that $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$0 is a $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$1-twisted Rota-Baxter operator if and only if $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$2 is a 3-Leibniz subalgebra of the $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$3-twisted semidirect product $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$4 (Teng, 30 Jul 2025). The same operator also induces a 3-Leibniz bracket on $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$5,

$$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$6

with respect to which $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$7 is a 3-Leibniz morphism: $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$8

The conceptual point is that NS-3-Leibniz algebras are not introduced independently of this operator formalism. They are the split structures naturally encoded by such operators, just as NS-Leibniz algebras are the underlying structures of twisted relative Rota-Baxter operators on ordinary Leibniz algebras (Das et al., 2021), and just as 3-tri-Leibniz algebras arise from embedding tensors on 3-Leibniz algebras (Teng et al., 6 Feb 2025).

3. Definition and axiomatic system

An NS-3-Leibniz algebra is a vector space $$[a,b,[x,y,z]_{\mathfrak g}]_{\mathfrak g} = [[a,b,x]_{\mathfrak g},y,z]_{\mathfrak g} + [x,[a,b,y]_{\mathfrak g},z]_{\mathfrak g} + [x,y,[a,b,z]_{\mathfrak g}]_{\mathfrak g}.$$9 equipped with four trilinear operations

$[\cdot,\cdot,\cdot]_{\vdash}$0

satisfying a system of six identities (Teng, 30 Jul 2025). Setting

$[\cdot,\cdot,\cdot]_{\vdash}$1

the defining relations are:

$[\cdot,\cdot,\cdot]_{\vdash}$2

$[\cdot,\cdot,\cdot]_{\vdash}$3

$[\cdot,\cdot,\cdot]_{\vdash}$4

$[\cdot,\cdot,\cdot]_{\vdash}$5

$[\cdot,\cdot,\cdot]_{\vdash}$6

$[\cdot,\cdot,\cdot]_{\vdash}$7

$[\cdot,\cdot,\cdot]_{\vdash}$8

These identities are exactly the axioms stated in the defining paper (Teng, 30 Jul 2025). They encode how the four pieces interact so that their sum behaves as a single 3-Leibniz bracket.

The notation strongly suggests a decomposition by “position” and “twist”: $[\cdot,\cdot,\cdot]_{\vdash}$9, $[\cdot,\cdot,\cdot]_{\dashv}$0, and $[\cdot,\cdot,\cdot]_{\dashv}$1 are the three representation-derived pieces, while $[\cdot,\cdot,\cdot]_{\dashv}$2 records the cocycle contribution. That interpretation is not merely heuristic; it is precisely how the four operations arise from a twisted Rota-Baxter operator, as discussed below (Teng, 30 Jul 2025).

4. Subadjacent 3-Leibniz bracket and reductions

A central structural fact is that

$[\cdot,\cdot,\cdot]_{\dashv}$3

is always a 3-Leibniz algebra (Teng, 30 Jul 2025). The operation $[\cdot,\cdot,\cdot]_{\dashv}$4 is therefore called the sub-adjacent 3-Leibniz bracket. In this sense, an NS-3-Leibniz algebra is a compatible refinement of an ordinary 3-Leibniz algebra.

Several reductions are explicitly recorded in the source (Teng, 30 Jul 2025).

• If $[\cdot,\cdot,\cdot]_{\dashv}$5, $[\cdot,\cdot,\cdot]_{\dashv}$6, and $[\cdot,\cdot,\cdot]_{\dashv}$7 are trivial, then $[\cdot,\cdot,\cdot]_{\dashv}$8 is a 3-Leibniz algebra.
• If $[\cdot,\cdot,\cdot]_{\dashv}$9, the structure reduces to the earlier notion of a 3-pre-Leibniz algebra.
• In the skew-symmetric case, NS-3-Leibniz becomes NS-3-Lie, recovering constructions of Hou–Sheng and Chtioui–Hajjaji–Mabrouk–Makhlouf.

These specializations identify NS-3-Leibniz algebras as part of a hierarchy of ternary algebraic splittings. They also clarify a common misconception: the theory is not simply a renaming of 3-Leibniz algebras. The point is not the presence of a single ternary Leibniz identity, but the existence of four operations constrained so that their sum is 3-Leibniz (Teng, 30 Jul 2025).

The binary precursor exhibits the same pattern. An NS-Leibniz algebra $[\cdot,\cdot,\cdot]_{\perp}$0 carries a total product

$[\cdot,\cdot,\cdot]_{\perp}$1

and the induced bracket $[\cdot,\cdot,\cdot]_{\perp}$2 is Leibniz (Das et al., 2021). NS-3-Leibniz algebras are the ternary analogue of this splitting principle.

5. Construction from twisted Rota-Baxter and related operators

Given a $[\cdot,\cdot,\cdot]_{\perp}$3-twisted Rota-Baxter operator $[\cdot,\cdot,\cdot]_{\perp}$4 on a 3-Leibniz algebra, the paper defines four trilinear operations on $[\cdot,\cdot,\cdot]_{\perp}$5 by

$[\cdot,\cdot,\cdot]_{\perp}$6

$[\cdot,\cdot,\cdot]_{\perp}$7

$[\cdot,\cdot,\cdot]_{\perp}$8

$[\cdot,\cdot,\cdot]_{\perp}$9

These operations satisfy the NS-3-Leibniz identities, so $\triangleright$0 becomes an NS-3-Leibniz algebra (Teng, 30 Jul 2025). Moreover, the associated sub-adjacent bracket is exactly the induced 3-Leibniz bracket: $\triangleright$1

The source also supplies three explicit operator families producing NS-3-Leibniz algebras (Teng, 30 Jul 2025).

Operator source	Resulting operations
Nijenhuis operator $\triangleright$2	$\triangleright$3, $\triangleright$4, $\triangleright$5, with the stated $\triangleright$6 formula
Reynolds operator $\triangleright$7	$\triangleright$8, $\triangleright$9, $\triangleleft$0, $\triangleleft$1
Rota-Baxter operator of weight $\triangleleft$2	$\triangleleft$3, $\triangleleft$4, $\triangleleft$5, with the stated weight-$\triangleleft$6 formula for $\triangleleft$7

For the Nijenhuis case, the fourth operation is

$\triangleleft$8

while for a Rota-Baxter operator $\triangleleft$9 of weight $\diamond$0,

$\diamond$1

These formulas are important because they show that NS-3-Leibniz structure is not purely formal; it is functorially generated by operator data (Teng, 30 Jul 2025).

A plausible implication is that NS-3-Leibniz algebras serve as a normal form for operator-induced splittings in the same way that 3-tri-Leibniz algebras do for embedding tensors (Teng et al., 6 Feb 2025), though the two frameworks are not identified in the cited sources.

6. Reconstruction, nearby frameworks, and deformation context

The defining paper proves a converse reconstruction statement: a 3-Leibniz algebra $\diamond$2 admits a compatible NS-3-Leibniz structure if and only if there exists an invertible $\diamond$3-twisted Rota-Baxter operator $\diamond$4 with respect to some representation $\diamond$5 and $\diamond$6-cocycle $\diamond$7 (Teng, 30 Jul 2025). In that case, the four operations are recovered by transport of structure: $\diamond$8

$\diamond$9

Thus invertible twisted Rota-Baxter operators and compatible NS-3-Leibniz structures are equivalent data.

The paper also places the theory into a deformation-theoretic setting. Twisted Rota-Baxter operators are Maurer-Cartan elements of an $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$0-algebra with only $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$1 and $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$2 nonzero, and the induced $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$3-differential $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$4 coincides with the coboundary operator $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$5 of the cohomology theory of twisted Rota-Baxter operators (Teng, 30 Jul 2025). Since NS-3-Leibniz algebras are the underlying structures of those operators, the theory is embedded in a homotopical and cohomological framework from the outset.

Several nearby constructions help situate the concept. NS-Leibniz algebras are the binary analogue, arising from $(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$6-twisted relative Rota-Baxter operators and splitting a Leibniz product into three operations (Das et al., 2021). 3-tri-Leibniz algebras arise from embedding tensors on 3-Leibniz algebras and can themselves be recovered from quotient embedding tensors (Teng et al., 6 Feb 2025). 3-Leibniz-Lie algebras add a ternary operation to a 3-Lie bracket and are induced by nonabelian embedding tensors on 3-Lie algebras (Teng et al., 2023). Crossed modules of ternary Leibniz algebras show that ternary Leibniz theories also support six-slot action systems and Peiffer-type identities (Gamou et al., 30 Jan 2025).

These related frameworks should not be conflated. The sources explicitly state that 3-Leibniz-Lie algebras are close in spirit to NS-3-Leibniz-type structures but do not use the same terminology or axiomatic system (Teng et al., 2023). Likewise, 3-tri-Leibniz algebras split ternary structure into three operations, not four (Teng et al., 6 Feb 2025). The distinctive feature of NS-3-Leibniz algebras is therefore the fourfold decomposition

$(\mathfrak g,[\cdot,\cdot,\cdot]_{\mathfrak g})$7

together with its precise realization by twisted Rota-Baxter data (Teng, 30 Jul 2025).

In summary, NS-3-Leibniz algebras are split ternary Leibniz structures whose defining significance is structural rather than merely formal: they are exactly the algebraic patterns produced by twisted Rota-Baxter operators on 3-Leibniz algebras, their sum always yields a sub-adjacent 3-Leibniz bracket, and compatible such splittings are equivalent to invertible twisted Rota-Baxter operators (Teng, 30 Jul 2025).