Dual Pre-Poisson Algebras Overview

• Dual pre-Poisson algebras are defined by a permutative product and a Leibniz bracket linked with compatibility identities that generalize classical Poisson algebras.
• Operadic duality shows that dual pre-Poisson algebras are the Koszul dual of pre-Poisson algebras, offering insights into deformation quantization and bialgebra frameworks.
• Diassociative deformations, O-operators, and Hom-algebra extensions illustrate their practical role in extending Yang–Baxter theory and noncommutative geometry.

A dual pre-Poisson algebra is an algebraic structure formed by a permutative algebra and a Leibniz algebra linked via compatibility conditions that generalize the usual Poisson algebra. This concept arises as the Koszul dual of the pre-Poisson algebra, which couples Zinbiel and pre-Lie structures; thus, dual pre-Poisson algebras naturally belong to the operadic duality context and serve as "dual" objects to pre-Poisson algebras in deformation theory and bialgebra theory (Lu, 16 Jan 2026).

1. Algebraic Structure and Defining Axioms

A dual pre-Poisson algebra is defined on a vector space $A$ equipped with two operations: a permutative product $\circ$ and a Leibniz bracket $[\,,\,]$. The permutative product satisfies

$x\circ(y\circ z) = (y \circ x)\circ z$

for all $x, y, z \in A$, encapsulating the essence of permutative operads. The Leibniz bracket satisfies

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

for all $x,y,z \in A$, adhering to the right Leibniz law.

These two operations are coupled by mixed identities: $[x, y\circ z] = [x, y]\circ z + y\circ[x, z]$

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

$$(x\circ y)\circ z = (y\circ x)\circ z,\qquad [x, y]\circ z = - [y,x]\circ z$$

which ensure full compatibility and guarantee that the construction generalizes the classical Poisson algebra (commutative associative product and a Lie bracket satisfying the Leibniz rule). These axioms define the precise categorical and operadic content of dual pre-Poisson algebras (Lu, 16 Jan 2026).

2. Operadic Duality and Structural Properties

In operad theory, the dual pre-Poisson operad $\mathsf{DualprePois}$ is the Koszul dual of the pre-Poisson operad $\mathsf{prePois}$, itself known to be Koszul. The dual pre-Poisson operad is thus Koszul, and its Hilbert–Poincaré series is

$H(t) = \frac{t}{(1-t)^2}$

indicating that

$\dim \mathsf{DualprePois}(n) = n \cdot n!$

for all $n$, reflecting maximal noncommutativity and intricate shuffle symmetries in the underlying tensor structure (Lu, 16 Jan 2026).

3. Deformation Quantizations and Semi-Classical Limits

A central methodology for constructing dual pre-Poisson algebras is via diassociative formal deformations of permutative algebras: $$x \triangleright_\hbar y = \sum_{i \geq 0} (x \triangleright_i y)\, \hbar^i$$

$$x \triangleleft_\hbar y = \sum_{i \geq 0} (x \triangleleft_i y)\, \hbar^i$$

giving rise to a dialgebra structure on $A[[\hbar]]$. The semi-classical limit (as $\hbar \to 0$) yields the dual pre-Poisson bracket

$$[x, y] = x \triangleright_1 y - y \triangleleft_1 x$$

and it can be systematically verified that the triple $(A, \circ, [\,,\,])$ satisfies all dual pre-Poisson axioms. This result establishes a direct algebraic correspondence between deformation quantization on dialgebras and dual pre-Poisson structures (Lu, 16 Jan 2026), analogous to how associative deformations yield pre-Poisson in the Koszul counterpart.

4. Dual Pre-Poisson Bialgebras, Manin Triples, and Matched Pairs

A dual pre-Poisson bialgebra is a sextuple $$(A, \circ, [\,,\,],\,\delta_{\circ}, \delta_{[\,,\,]})$$ where the coalgebra structures $\delta_{\circ}$ (permutative) and $\delta_{[\,,\,]}$ (Leibniz) intermesh via dualized versions of the algebraic compatibility relations. Three equivalent characterizations arise:

• Manin triples: The quadratic dual pre-Poisson algebra $(A \oplus A^*, \circ, [\,,\,], B)$, with $$B(x+\alpha, y+\beta) = \langle x, \beta \rangle - \langle y, \alpha \rangle$$, and $A$ and $A^*$ isotropic subalgebras.
• Matched pairs: Two dual pre-Poisson algebras $A_1, A_2$ with eight linear maps $l_\circ, r_\circ, l_{[\,,\,]}, r_{[\,,\,]}$ acting mutually and satisfying sixteen quadratic relations; the bicrossed product $A_1 \bowtie A_2$ then forms the dual pre-Poisson algebra.
• Coproduct definition: Explicit compatibility relations between the coalgebra maps $\delta_{\circ}$, $\delta_{[\,,\,]}$ and the algebra structure (Lu, 16 Jan 2026).

This equivalence gives rise to a robust framework for generalizing bialgebra theory beyond the classical (Lie, associative, Poisson) context.

5. The Permutative–Leibniz Yang–Baxter Equation (PLYBE)

Dual pre-Poisson bialgebras admit a Yang–Baxter theory governed by the permutative–Leibniz Yang–Baxter equation. For $r = \sum_i a_i \otimes b_i$,

$$P(r) = r_{13}\circ r_{23} - r_{12} \circ r_{23} + r_{13} \blacksquare r_{12}$$

$$L(r) = [r_{13}, r_{23}] + [r_{12}, r_{23}] - r_{12} \square r_{13}$$

with $x \blacksquare y = x \circ y - y \circ x$ and $x \square y = [x,y] + [y,x]$. Symmetric solutions $r$ to $P(r) = 0$, $L(r) = 0$ systematically yield dual pre-Poisson bialgebra structures, paralleling the role of the classical Yang–Baxter equation in Lie bialgebra theory (Lu, 16 Jan 2026).

6. $\mathcal{O}$-Operators and Pre-Dual Pre-Poisson Algebras

An $\mathcal{O}$-operator for a dual pre-Poisson algebra $(A, \circ, [\,,\,])$ with respect to a module $(V; l_\circ, r_\circ, l_{[\,,\,]}, r_{[\,,\,]})$ is a map $T: V \to A$ satisfying

$$T(u)\circ T(v) = T(l_\circ(T(u))v + r_\circ(T(v))u)$$

$$[T(u),T(v)] = T(l_{[\,,\,]}(T(u))v + r_{[\,,\,]}(T(v))u)$$

Such operators induce a pre-dual pre-Poisson structure on $V$ via

$$u\rhd_{V} v = l_\circ(T(u))(v), \quad u\lhd_{V} v = r_\circ(T(v))(u)$$

$$u\succ_{V} v = l_{[\,,\,]}(T(u))(v), \quad u\prec_{V} v = r_{[\,,\,]}(T(v))(u)$$

Splittings via Rota–Baxter operators induce pre-dual pre-Poisson structures, and invertible $\mathcal{O}$-operators yield canonical dual structures.

Examples include the canonical 2-dimensional case with $e_2 \circ e_2 = [e_2, e_2] = e_1$, exhibiting a nontrivial Rota–Baxter splitting, and semidirect sums presenting explicit symmetric solutions to the PLYBE. This provides a systematic construction for dual pre-Poisson bialgebras from representation-theoretic data (Lu, 16 Jan 2026).

7. Dual–Hom–Pre–Poisson Extensions

In Hom-algebraic settings, a dual–Hom–pre–Poisson algebra $(A, \cdot, \{\,,\,\}, \alpha)$ imposes Hom-permutativity, Hom-Leibniz, and Hom-derivation among its operations and a twisting linear map $\alpha$. The compatibility inherits all defining relations with each variable multiplied by an application of $\alpha$ or $\alpha^{-1}$, mirroring the classical dual pre-Poisson but generalizing to Hom-structures (Liu et al., 2021).

Such algebras can be constructed from Hom–Poisson algebras equipped with a Hom-average-operator, leading to new Hom-dual-pre-Poisson structures and functorial connections between Hom–Poisson and dual–Hom–pre–Poisson categories. These objects capture inherently twisted deformation-theoretic phenomena.

Summary Table: Core Dual Pre-Poisson Structures

Structure	Defining Operation(s)	Key Identity/Compatibility
Permutative algebra	$\circ$	$x\circ(y\circ z) = (y\circ x)\circ z$
Leibniz algebra	$[\,,\,]$	$[x,[y,z]]= [[x,y],z]+[y,[x,z]]$
Dual pre-Poisson algebra	$\circ$, $[\,,\,]$	(mixed compatibilities above)
Dual pre-Poisson bialg.	$+,(\delta_\circ,\delta_{[\,,\,]})$	(bialgebra & Manin triple equivalence)
PLYBE (Yang–Baxter)	$r$	$P(r)=0,\ L(r)=0$

Dual pre-Poisson algebras and bialgebras provide a foundational algebraic formalism for addressing generalized Yang–Baxter equations, deformation quantizations, and bialgebra structures beyond the classical Lie and Poisson realms, with significant implications in operad theory, quantum algebra, and noncommutative geometry (Lu, 16 Jan 2026, Liu et al., 2021).