Almost Tridendriform Poisson Algebras

• The paper demonstrates that almost tridendriform Poisson algebras split the commutative product and Leibniz bracket, reconstructing an almost Poisson algebra via sub-adjacent operations.
• Their structure builds on commutative dendriform trialgebra principles, using a pre-bracket and dendriform splitting to yield a coherent algebraic system.
• Weighted relative Rota–Baxter operators facilitate the construction of ATdP algebras, embedding them in AWB frameworks and linking them to almost Poisson D-bialgebra theory.

Searching arXiv for the cited paper to ground the article and verify metadata. Almost tridendriform Poisson algebras are algebraic structures introduced in "D-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras" (Mabrouk, 20 Mar 2026) as a dendrified form of almost Poisson algebras. They combine a commutative associative product, a skew-symmetric Leibniz-type bracket, a dendriform splitting of the product, and a pre-bracket operation whose antisymmetrization contributes to the induced almost Poisson bracket. The defining point is that the split operations reconstruct an associated almost Poisson algebra through a sub-adjacent commutative product and an associated skew-symmetric bracket, while also arising canonically from weighted relative Rota–Baxter operators (Mabrouk, 20 Mar 2026).

1. Definition within the almost Poisson and AWB framework

An almost Poisson algebra is a triple $(A,[\cdot,\cdot],\cdot)$ in which $(A,\cdot)$ is a commutative associative algebra, $[\cdot,\cdot]$ is a bilinear skew-symmetric bracket, and the bracket satisfies the Leibniz rule

$[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$

for all $x,y,z\in A$. No Jacobi identity is assumed, so the bracket need not be a Lie bracket (Mabrouk, 20 Mar 2026).

This places almost Poisson algebras in direct relation to algebras with bracket (AWB). An AWB is a triple $(A,\cdot,\{\cdot,\cdot\})$ where $(A,\cdot)$ is associative, not necessarily commutative, and the bracket satisfies the left Leibniz-type compatibility

$\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$

When the product is commutative and the bracket is skew-symmetric, an AWB reduces to an almost Poisson algebra (Mabrouk, 20 Mar 2026).

The notion of almost tridendriform Poisson algebra is built precisely in this relaxed Leibniz setting. The qualifier “almost” indicates that the Jacobi identity is not part of the definition. This distinguishes the structure from genuinely Poisson variants while preserving the derivation property that governs interaction between bracket and product (Mabrouk, 20 Mar 2026).

2. Algebraic structure and defining identities

The construction begins with a commutative dendriform trialgebra in the two-operation presentation used in the paper. This is a vector space $A$ equipped with bilinear operations $\cdot$ and $(A,\cdot)$0 such that $(A,\cdot)$1 is a commutative associative algebra and

$(A,\cdot)$2

for all $(A,\cdot)$3 (Mabrouk, 20 Mar 2026).

These identities ensure that the sub-adjacent product

$(A,\cdot)$4

is commutative and associative. This sub-adjacent algebra is denoted $(A,\cdot)$5 in the paper (Mabrouk, 20 Mar 2026).

An almost tridendriform Poisson algebra is then a quintuple

$(A,\cdot)$6

such that:

1. $(A,\cdot)$7 is an almost Poisson algebra;
2. $(A,\cdot)$8 is a commutative dendriform trialgebra;
3. with

$(A,\cdot)$9

the following compatibilities hold for all $[\cdot,\cdot]$0:

$[\cdot,\cdot]$1

$[\cdot,\cdot]$2

$[\cdot,\cdot]$3

$[\cdot,\cdot]$4

These relations encode the simultaneous splitting of the commutative product and the bracket (Mabrouk, 20 Mar 2026).

A useful interpretation is that $[\cdot,\cdot]$5 controls the dendriform decomposition of the product, while $[\cdot,\cdot]$6 acts as a pre-bracket whose skew-symmetrization, together with the original almost Poisson bracket, produces the associated bracket $[\cdot,\cdot]$7. This suggests a structural analogy with other splitting constructions in operadic algebra, although the paper states the result specifically in the almost Poisson setting (Mabrouk, 20 Mar 2026).

3. Passage to the associated almost Poisson algebra

A central theorem states that every almost tridendriform Poisson algebra canonically determines an almost Poisson algebra. For an ATdP algebra $[\cdot,\cdot]$8, the sub-adjacent product and associated bracket are

$[\cdot,\cdot]$9

Then $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$0 is a commutative associative algebra, $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$1 is skew-symmetric, and the Leibniz rule

$[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$2

holds for all $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$3 (Mabrouk, 20 Mar 2026).

The proof strategy described in the paper is an expansion of the defining identities (ATdP-1)–(ATdP-4), together with the dendriform-to-associative reduction for $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$4 (Mabrouk, 20 Mar 2026). In this sense, ATdP structures are not merely enrichments of almost Poisson algebras; they are splittings whose recombination is guaranteed to recover an almost Poisson algebra.

The paper also proves that an ATdP algebra supplies canonical module data over its associated almost Poisson algebra $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$5. Specifically, the left actions

$[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$6

make $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$7 into an $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$8-module almost Poisson algebra (Mabrouk, 20 Mar 2026). This result situates the internal ATdP operations as representation-theoretic data for the associated algebra.

4. Construction from weighted relative Rota–Baxter operators

The main construction of almost tridendriform Poisson algebras is via weighted relative Rota–Baxter operators. Let $[x,y\cdot z]=[x,y]\cdot z+y\cdot [x,z],$9 be an almost Poisson algebra, and let $x,y,z\in A$0 be an $x,y,z\in A$1-module almost Poisson algebra. Here $x,y,z\in A$2 is a representation of the associative algebra $x,y,z\in A$3 satisfying

$x,y,z\in A$4

and the representation data obey

$x,y,z\in A$5

If $x,y,z\in A$6 itself carries an almost Poisson structure, the module compatibilities are

$x,y,z\in A$7

and

$x,y,z\in A$8

(Mabrouk, 20 Mar 2026).

A linear map $x,y,z\in A$9 is a weighted relative Rota–Baxter operator of weight $(A,\cdot,\{\cdot,\cdot\})$0 if

$(A,\cdot,\{\cdot,\cdot\})$1

and, in the almost Poisson setting,

$(A,\cdot,\{\cdot,\cdot\})$2

for all $(A,\cdot,\{\cdot,\cdot\})$3 (Mabrouk, 20 Mar 2026).

From such an operator the paper derives ATdP structure in two steps. First, the associative identity yields a commutative dendriform trialgebra on $(A,\cdot,\{\cdot,\cdot\})$4 by

$(A,\cdot,\{\cdot,\cdot\})$5

Second, after including the bracket identity, one defines

$(A,\cdot,\{\cdot,\cdot\})$6

The resulting structure $(A,\cdot,\{\cdot,\cdot\})$7 is an almost tridendriform Poisson algebra (Mabrouk, 20 Mar 2026).

The same theorem states that the operator $(A,\cdot,\{\cdot,\cdot\})$8 is a homomorphism from the associated almost Poisson algebra $(A,\cdot,\{\cdot,\cdot\})$9 to $(A,\cdot)$0:

$(A,\cdot)$1

This identifies ATdP algebras as dendrifications of almost Poisson algebras arising from relative Rota–Baxter operators (Mabrouk, 20 Mar 2026).

5. Canonical examples and special cases

A distinguished special case occurs when $(A,\cdot)$2, $(A,\cdot)$3 is left multiplication, and $(A,\cdot)$4 is the adjoint action for the bracket. Then a weighted Rota–Baxter operator $(A,\cdot)$5 of weight $(A,\cdot)$6 on the almost Poisson algebra $(A,\cdot)$7 yields the ATdP operations

$(A,\cdot)$8

(Mabrouk, 20 Mar 2026).

The paper gives a concrete example with $(A,\cdot)$9 and $\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$0. In that case the Rota–Baxter identities hold both for the commutative product and the bracket, and the induced operations become

$\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$1

The sub-adjacent product is

$\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$2

because the product is commutative, and the associated almost Poisson bracket is $\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$3, which still satisfies the Leibniz rule because $\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$4 does (Mabrouk, 20 Mar 2026).

The paper also verifies one of the ATdP axioms explicitly in this example. Condition (ATdP-4),

$\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$5

reduces to

$\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$6

which is exactly the Leibniz rule for the original bracket (Mabrouk, 20 Mar 2026). This example clarifies that the ATdP formalism does not require exotic input; even the identity map at weight $\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$7 produces a nontrivial splitting.

6. Embeddings into AWB and relation to D-bialgebra theory

The paper places almost tridendriform Poisson algebras alongside two broader constructions: embeddings into AWB and almost Poisson $\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$8-bialgebras (Mabrouk, 20 Mar 2026).

For embeddings, let $\{x,y\cdot z\}=\{x,y\}\cdot z+y\cdot \{x,z\}.$9 be an almost Poisson algebra and $A$0 a representation. The hemisemi-direct sum $A$1 becomes an AWB, denoted $A$2, with product and bracket

$A$3

$A$4

for all $A$5 and $A$6 (Mabrouk, 20 Mar 2026).

If $A$7 is a relative averaging operator for $A$8 with respect to $A$9, then the graph

$\cdot$0

is an AWB subalgebra. The graph characterization theorem states that $\cdot$1 is a relative averaging operator if and only if $\cdot$2 is an AWB subalgebra (Mabrouk, 20 Mar 2026). In the case $\cdot$3 with the adjoint representation, an averaging operator embeds the almost Poisson algebra into an AWB and induces operations

$\cdot$4

The $\cdot$5-bialgebra direction is conceptually parallel but structurally distinct. The paper defines almost Poisson $\cdot$6-bialgebras as objects $\cdot$7 combining an almost Poisson algebra structure, an almost Poisson coalgebra structure, a commutative infinitesimal $\cdot$8-bialgebra structure, and additional compatibility axioms (Mabrouk, 20 Mar 2026). It proves the equivalence between:

Object	Equivalent formulation
Almost Poisson $\cdot$9-bialgebra	Matched pair
Almost Poisson $(A,\cdot)$00-bialgebra	Standard Manin triple
Matched pair	Standard Manin triple

More precisely, the equivalence is between almost Poisson $(A,\cdot)$01-bialgebras, matched pairs $(A,\cdot)$02, and standard Manin triples $(A,\cdot)$03 with canonical pairing

$(A,\cdot)$04

invariant for both operations (Mabrouk, 20 Mar 2026).

The paper explicitly presents both $(A,\cdot)$05-bialgebras and ATdP algebras as manifestations of “splitting” or “doubling” principles: $(A,\cdot)$06-bialgebras through matched pairs and Manin triples, and ATdP algebras through relative Rota–Baxter dendrification, where both the product and the bracket are decomposed into pre-operations (Mabrouk, 20 Mar 2026). A plausible implication is that ATdP theory should be read not as an isolated definition, but as one component of a broader program for almost Poisson analogues of classical Poisson and Lie-theoretic constructions.

7. Scope, conventions, and conceptual significance

The framework is formulated over a field $(A,\cdot)$07 of characteristic $(A,\cdot)$08, and all vector spaces and algebras under consideration are finite-dimensional. The paper uses $(A,\cdot)$09 for the tensor flip, Sweedler notation for coproducts and cobrackets, $(A,\cdot)$10 for left multiplication, and $(A,\cdot)$11 for the left action induced by the bracket (Mabrouk, 20 Mar 2026).

Within this setting, almost tridendriform Poisson algebras serve as the split counterparts of almost Poisson algebras. The product splitting is encoded by $(A,\cdot)$12 together with the commutative associative operation $(A,\cdot)$13, while the bracket splitting is encoded by $(A,\cdot)$14 together with the original bracket $(A,\cdot)$15. Their recombination yields the associated operations $(A,\cdot)$16 and $(A,\cdot)$17, which satisfy the almost Poisson axioms (Mabrouk, 20 Mar 2026).

A common misconception would be to treat “almost” as a minor technical caveat. In fact, it marks a substantive relaxation: the Jacobi identity is absent throughout the ATdP definition and throughout the underlying almost Poisson theory. What remains essential is the Leibniz derivation property of the bracket with respect to the product (Mabrouk, 20 Mar 2026). Another possible misunderstanding is to identify ATdP algebras with arbitrary product splittings; the paper’s results are sharper than that, since the principal characterization ties them specifically to weighted relative Rota–Baxter operators and module almost Poisson data (Mabrouk, 20 Mar 2026).

In the formulation developed there, ATdP algebras occupy a precise place among noncommutative and Leibniz-type generalizations of Poisson structures. They arise from operator-theoretic input, produce associated almost Poisson algebras functorially through sub-adjacent operations, and coexist with a parallel Drinfel'd-type theory of almost Poisson $(A,\cdot)$18-bialgebras, matched pairs, and Manin triples (Mabrouk, 20 Mar 2026).