RPCA Yang–Baxter Equation

• RPCA–YBE is a specialized form of the Yang–Baxter equation that operates within relative PCA algebras, encoding both associative and anti–pre–Lie operations with differential compatibility.
• It synthesizes constraints from associative multiplication and anti–pre–Lie operations to generate coboundary bialgebra structures used in quantization, deformation, and integrable model construction.
• The theory leverages O–operators and split relative pre–PCA algebras to construct explicit antisymmetric tensor solutions, addressing limitations of classical bialgebra approaches.

The Relative PCA Yang–Baxter Equation (RPCA–YBE) is a specialized form of the Yang–Baxter equation formulated within the context of relative PCA algebras—algebraic structures designed to encode the bialgebraic features of relative Poisson algebras and their differential extensions. The RPCA–YBE synthesizes compatibility conditions from both associative and anti–pre–Lie algebraic operations, enhanced by the presence of derivations, in order to generate coboundary bialgebra structures that are suitable for quantization, deformation, and the paper of integrable systems.

1. Algebraic Foundation: Relative PCA Algebras

A relative PCA algebra $(A, \cdot, \circ, P, Q)$ consists of a commutative associative multiplication “$\cdot$”, an anti–pre–Lie product “$\circ$”, and linear maps $P, Q: A \to A$ that generalize differential operators. The algebraic structure generalizes the interplay present in relative Poisson algebras—where the commutative product and Lie bracket are related through modified Leibniz rules. In relative PCA algebras, the anti–pre–Lie operation $\circ$ subsumes the Lie bracket as its commutator, and the maps $P, Q$ encode additional structural variation beyond classical invariance, specifically by involving a commutative 2–cocycle on the sub-adjacent Lie algebra (Liu et al., 14 Sep 2025).

2. Mathematical Formulation of RPCA–YBE

Within this setting, the RPCA–YBE is the condition on a tensor $r \in A \otimes A$ (typically antisymmetric: $r = -\tau(r)$ under the flip map) defined by the vanishing of two distinct algebraic expressions:

• Associative Yang–Baxter term:

$$A(r) = r_{12} \cdot r_{13} - r_{23} \cdot r_{12} + r_{13} \cdot r_{23} = 0$$

• Anti–Pre–Lie Yang–Baxter term:

$$T(r) = r_{12} \circ r_{13} + r_{12} \circ r_{23} - [r_{13}, r_{23}] = 0$$

where $[x, y] = x \circ y - y \circ x$.

• Differential compatibility:

$$(Q \otimes \mathrm{id} - \mathrm{id} \otimes P) r = 0$$

A solution $r$ of these conditions—especially in the antisymmetric case—provides the structure necessary to define coboundary relative PCA bialgebras, with explicit comultiplication maps given by

$$\Delta(x) = (\mathrm{id} \otimes \mathcal{L}_\cdot(x) - \mathcal{L}_\cdot(x) \otimes \mathrm{id}) r,$$

$$\theta(x) = (\mathcal{L}_\circ(x) \otimes \mathrm{id} - \mathrm{id} \otimes \mathrm{ad}(x)) r,$$

for all $x \in A$, where $\mathcal{L}_\cdot(x)$ is left-multiplication in the associative algebra and $\mathrm{ad}(x)$ is the adjoint action in the anti–pre–Lie algebra (Liu et al., 14 Sep 2025).

3. Analytical and Combinatorial Tools: $\mathcal{O}$–Operators and Relative Pre–PCA Algebras

To construct solutions of the RPCA–YBE, the paper introduces the notion of $\mathcal{O}$–operators. Given a representation of the relative PCA algebra

$$(\mu,\, l_\circ,\, r_\circ,\, \alpha,\, \beta,\, V),$$

a linear operator $T: V \to A$ is an $\mathcal{O}$–operator if it satisfies:

• For the associative structure:

$T(u) \cdot T(v) = T(\mu(T(u)) v + \mu(T(v)) u),$

• For the anti–pre–Lie structure:

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

• Compatibility with derivations:

$$P \circ T = T \circ \alpha, \quad Q \circ T = T \circ \beta.$$

When $r = T - \tau(T)$ is antisymmetric, it is a solution to the RPCA–YBE (Liu et al., 14 Sep 2025).

Relative pre–PCA algebras serve as “split” forms, in which the anti–pre–Lie product is decomposed into left and right operations: $\circ = \succ + \prec$. The identity map in this context naturally gives $\mathcal{O}$–operators that yield solutions of the RPCA–YBE.

4. Coboundary Relative PCA Bialgebras and Their Significance

Antisymmetric solutions $r$ of the RPCA–YBE produce coboundary relative PCA bialgebras—a generalization of the familiar coboundary Lie bialgebras from the classical Yang–Baxter context. These bialgebras have comultiplications $\Delta, \theta$ as specified above, and serve as the algebraic foundation for applications in deformation theory, quantization, and the paper of integrable systems with a relative or differential structure.

This approach offers an alternative to the traditional Manin triple/invariance-based bialgebra theory, replacing invariance by 2–cocycle conditions, and yielding structures compatible with commutative and cocommutative differential antisymmetric infinitesimal (ASI) bialgebras (Liu et al., 14 Sep 2025).

5. Comparative Context: RPCA–YBE Versus Classical YBEs

The RPCA–YBE unifies and generalizes several prior forms of the Yang–Baxter equation:

• In classical Lie algebras, the classical Yang–Baxter equation generates Lie bialgebras via suitable antisymmetric solutions.
• In associative algebras, the associative Yang–Baxter equation gives rise to infinitesimal bialgebra structures.
• Anti–pre–Lie Yang–Baxter equations yield anti–pre–Lie bialgebras.

In a relative PCA algebra, both associative and anti–pre–Lie components are simultaneously present, with the RPCA–YBE imposing intertwined constraints on the tensor $r$. The presence of differential operators $P$ and $Q$ introduces new compatibility conditions not found in the classical setting, reflecting the requirement for a bialgebra theory compatible with differential ASI bialgebras arising from commutative differential algebraic contexts (Liu et al., 14 Sep 2025).

6. Applications and Implications

The RPCA–YBE and its associated bialgebraic theory enable several novel constructions:

• Systematic generation of coboundary relative PCA bialgebras, suitable for quantization and the development of Frobenius Jacobi algebras.
• Direct integration with the theory of commutative and cocommutative differential ASI bialgebras, permitting the construction of quantized or deformed algebraic structures in settings (such as mathematical physics or geometry) where differential and relative structures coexist.
• A pathway to constructing and classifying new types of integrable models where classical invariance or standard Manin triple approaches do not suffice.

This framework responds to incompatibilities between standard Poisson bialgebra theories and differential bialgebra structures, offering a theory that is consistent with the latter via commutative 2–cocycles, relative split products, and $\mathcal{O}$–operator constructions (Liu et al., 14 Sep 2025).

7. Concluding Remarks

The Relative PCA Yang–Baxter Equation (RPCA–YBE) represents an important extension of the classical Yang–Baxter paradigm, tailored to the algebraic landscape of relative PCA algebras and the needs of differential (ASI) bialgebra theory. By requiring antisymmetric tensors to satisfy entwined associative and anti–pre–Lie compatibility relations—plus differential coherence—the RPCA–YBE robustly generates coboundary bialgebras suited to quantized, deformed, or differential applications. The constructive machinery of $\mathcal{O}$–operators and the embedding of relative pre–PCA algebras further broaden the scope for explicit solutions and integrable models within this expanded algebraic setting (Liu et al., 14 Sep 2025).