Relative PCA Bialgebras

• Relative PCA bialgebras are algebraic structures that extend traditional bialgebras by incorporating differential operators, combinatorial elements, and anti-pre-Lie operations.
• They achieve compatibility using dual coproducts and Manin triple analogs, unifying associative, Lie-type, and Poisson frameworks within a single structure.
• Their applications span deformation theory, quantum algebra, and integrable systems, providing robust tools for advanced algebraic analysis.

Relative PCA bialgebras constitute a recently systematized class of algebraic structures that extend the theory of classical bialgebras, Poisson bialgebras, and their relative generalizations. The term “PCA” (Principal Combinatorial Algebra or Poisson–Combinatorial–Algebraic, depending on context) encapsulates the presence of jointly associative and Lie-type operations, defined in a manner “relative” to additional algebraic data—either combinatorial (e.g. permutations), module-theoretic (e.g. derivations or auxiliary subalgebras), or governed by specific compatibility conditions. Contemporary research has clarified their equivalence to certain matched pair and Manin triple constructions, introduced non-classical Yang–Baxter equations, and established their foundational role in the paper of differential ASI bialgebras, Malcev–Poisson algebras, and quantum algebraic structures.

1. Definition and Fundamental Structure

A relative PCA bialgebra is an augmentation of the classical bialgebra axioms informed by differential, combinatorial, and duality-theoretic considerations. Let $A$ be a commutative associative algebra equipped with a derivation $P\colon A\to A$ (often arising from a differential algebra structure) (Liu et al., 14 Sep 2025). One introduces a second linear operator $Q$ and defines an “anti-pre-Lie” product via: $x \circ y = Q(xy) - P(x)y,\quad \forall x,y \in A.$ The quadruple $(A,\cdot,\circ,P,Q)$ is thus termed a relative PCA algebra.

For the bialgebra structure, one defines two compatible coproducts:

• $\Delta\colon A \rightarrow A\otimes A$, the associative coalgebra part, is required to be commutative and cocommutative.
• $\theta\colon A \rightarrow A\otimes A$, the anti-pre-Lie coalgebra part, is given by: $$\theta(x) = \Delta(P(x)) - (Q \otimes \mathrm{id})\Delta(x), \quad x \in A.$$ The axioms governing $(A,\cdot,\circ,\Delta,\theta,P,Q)$ ensure compatibility conditions linking the associative and anti-pre-Lie parts through commutators, Leibniz-type rules, and cohomological identities. Such objects generalize both relative Poisson bialgebras (Liu et al., 2023, Liu et al., 14 Sep 2025) and Malcev–Poisson bialgebras (Harrathi et al., 27 Feb 2025).

2. Manin Triples, Matched Pairs, and Invariance Conditions

Manin triple theory is a standard framework for encoding bialgebraic structures via the geometric data of a double algebra equipped with a symmetric bilinear form (Liu et al., 2023, Liu et al., 14 Sep 2025). In the context of relative PCA bialgebras, the “double” space $A\oplus A^*$ acquires the Witt-type symmetric form: $$\mathcal{B}_d(x+a^*, y+b^*) = \langle x,b^* \rangle + \langle a^*,y \rangle.$$ Classically, Manin triples rely on the invariance of this form under both product and bracket. For relative PCA bialgebras, the invariance is replaced with the requirement that $\mathcal{B}_d$ is a commutative $2$-cocycle: $$\mathcal{B}_d([x,y],z) + \mathcal{B}_d([y,z],x) + \mathcal{B}_d([z,x],y) = 0.$$ This relaxation is necessary for consistency with the differential and anti-pre-Lie structure, especially when the bracket arises via the Witt construction (e.g. $[x,y]=x P(y) - P(x)y$).

The underlying categorical equivalences have been established as follows:

• Relative PCA bialgebras $\longleftrightarrow$ matched pairs of relative Poisson (or Malcev–Poisson) algebras (including actions via representations and dual derivations),
• Relative PCA bialgebras $\longleftrightarrow$ Manin triples with commutative $2$-cocycle forms (Liu et al., 14 Sep 2025, Harrathi et al., 27 Feb 2025).

3. Coboundary Structures and Relative PCA Yang–Baxter Equations

In analogy with the classical associative and Lie bialgebra theory, relative PCA bialgebras feature coboundary constructions governed by generalized Yang–Baxter equations:

• Given an antisymmetric element $r \in A\otimes A$, define the coproducts: $$\Delta(x) = (\mathrm{id} \otimes L_\cdot(x) - L_\cdot(x) \otimes \mathrm{id})\, r, \quad \theta(x) = (\mathcal{L}_\circ(x)\otimes \mathrm{id} - \mathrm{id}\otimes \mathrm{ad}(x))\, r.$$
• The RPCA–YBE (relative PCA Yang–Baxter equation) comprises three conditions (Liu et al., 14 Sep 2025):
  • Associative YBE: $$A(r)= r_{12}\cdot r_{13} - r_{23}\cdot r_{12} + r_{13}\cdot r_{23}= 0$$,
  • Anti-pre-Lie YBE: $$T(r) = r_{12}\circ r_{13} + r_{12}\circ r_{23} - [r_{13}, r_{23}] = 0$$,
  • Operators $P, Q$ must satisfy: $$(P\otimes\mathrm{id} - \mathrm{id}\otimes Q)\,r = 0$$.

Antisymmetric solutions $r$ yield coboundary relative PCA bialgebras. This procedure unifies the coalgebraic part with the differential and anti-pre-Lie algebraic structures, thus extending the classical r-matrix formalism.

4. $\mathcal{O}$-Operators and Relative Pre-PCA Algebras

$\mathcal{O}$-operators generalize Rota–Baxter and pre-Lie operators to the relative PCA setting. Given a representation $(V, \mu, l_\circ, r_\circ)$ of a relative PCA algebra and structure maps $\alpha, \beta$ on $V$ corresponding to the derivations, a linear operator $T: V\to A$ is called an $\mathcal{O}$-operator if \begin{align*} T(u)\cdot T(v) &= T\big(\mu(T(u))v + \mu(T(v))u\big), \ T(u)\circ T(v) &= T\big(l_\circ(T(u))v + r_\circ(T(v))u\big), \ P \circ T &= T\circ \alpha, \quad Q\circ T = T\circ \beta. \end{align*} Antisymmetrization ($r = T - \tau(T)$) delivers solutions of the RPCA–YBE in the semidirect product algebra, thereby defining coboundary relative PCA bialgebras (Liu et al., 14 Sep 2025).

Relative pre-PCA algebras, equipped with a differential Zinbiel structure and an admissible splitting, serve as sources for such operators. The identity map in a pre-PCA algebra is always an $\mathcal{O}$-operator, trivially yielding antisymmetric solutions.

5. Connections to Other Bialgebraic Frameworks

Relative PCA bialgebras generalize several distinct strands within bialgebra theory:

• Relative Poisson bialgebras and Jacobi/Frobenius Jacobi algebras: PCA bialgebras resolve issues with unit elements and extend matched pairs and Manin triple constructions to unital cases via relaxation to $2$-cocycle forms (Liu et al., 2023, Liu et al., 14 Sep 2025).
• Malcev–Poisson bialgebras: These employ non-associative Malcev brackets with compatible associative products linked by the Leibniz rule. Post–Malcev–Poisson algebras and weighted relative Rota–Baxter operators underlie more general relative PCA bialgebra constructions in this setting (Harrathi et al., 27 Feb 2025).
• PROP-theoretic combinatorial models: The combinatorial PROP construction furnishes a normal form for every morphism, using explicit permutation data that characterize relative PCA bialgebras with noncommutativity controlled by combinatorial orderings (Becerra, 2021).
• Lie-theoretic correspondences in characteristic zero: The equivalence between categories of generalized-primitively-generated bialgebras and suspensive Lie algebras, as in generalized Milnor–Moore theorems, elucidates the structure of relative PCA bialgebras with noninvertible grouplikes (Beauvais-Feisthauer et al., 2022).

6. Applications and Research Directions

Relative PCA bialgebras have emerged as central structures in the theory of deformation and quantization, especially where classical Poisson or bialgebra symmetries are “split” by external or non-invertible algebraic data, such as in quantum groups, algebraic topology, and integrable systems (Lazarev et al., 2020, Liu et al., 14 Sep 2025).

They provide alternative frameworks for:

• Simultaneous deformations of associative and Lie-type structures via Maurer–Cartan functorial approaches (Lazarev et al., 2020).
• Computational and algorithmic models in symbolic algebra, leveraging the explicit normal forms and permutation data (Becerra, 2021).
• The construction of coboundary and quantized analogs of bialgebras where classical unit and symmetry assumptions fail, notably in the context of non-Hopf topology and moduli (Beauvais-Feisthauer et al., 2022, Harrathi et al., 27 Feb 2025).
• Bridging commutative differential algebra theory and noncommutative combinatorial models, particularly for modules with derivations and representation theory (Liu et al., 14 Sep 2025).

Research continues toward:

• Classification in various dimensions and characteristics,
• Quantization of coboundary structures and analytic parametrization of solutions to RPCA–YBE,
• Extension to super and graded settings, and applications in mathematical physics,
• Deepening connections to category-theoretic and homotopical algebra frameworks via functorial MC equations and enriched deformation theory (Lazarev et al., 2020, Liu et al., 14 Sep 2025).

7. Summary Table of Central Concepts

Structure/Class	Defining Data/Axioms	Key Reference
Relative PCA Bialgebra	$(A,\cdot,\circ,\Delta,\theta,P,Q)$; anti-pre-Lie splitting, $2$-cocycle Manin triple	(Liu et al., 14 Sep 2025)
RPCA Yang–Baxter Equation (YBE)	Associative YBE, anti-pre-Lie YBE, $P,Q$-compatibility; antisymmetric $r$	(Liu et al., 14 Sep 2025)
Manin triple (Witt form)	$A\oplus A^*$, symmetric bilinear form, commutative $2$-cocycle invariance	(Liu et al., 14 Sep 2025)
$\mathcal{O}$-operator	Representation, operator equations, derivation compatibility	(Liu et al., 14 Sep 2025)
Malcev–Poisson bialgebra	Non-associative bracket + associative product, compatible co-structure	(Harrathi et al., 27 Feb 2025)
Combinatorial PROP Extension	Morphisms with permutation data, normal form decomposition	(Becerra, 2021)

Relative PCA bialgebras thus embody a robust algebraic paradigm integrating differential, combinatorial, and categorical techniques, facilitating the paper and explicit construction of bialgebraic structures beyond those governed by classical or Lie-theoretic symmetries.