Special Apre-Perm Bialgebra Overview

• Special Apre-Perm bialgebras are bialgebraic structures featuring a distinctive two-part splitting of perm algebra operations with a commutative component.
• They extend averaging operators on commutative algebras by inducing a dual multiplication that interlinks with Yang–Baxter and Rota–Baxter operator theories.
• They establish structural links to Manin triples and Frobenius commutative algebras, enabling rigorous operator and double construction analyses.

A special Apre-Perm bialgebra is a bialgebraic structure that encapsulates a novel symmetry-refined splitting of the classical perm algebra operations, induced from averaging commutative and cocommutative infinitesimal bialgebras. Developed to extend the established process by which averaging operators on commutative algebras generate perm algebras, special Apre-Perm bialgebras are characterized by a two-part decomposition of the perm product, compatibility with left-invariant symmetric bilinear forms, and structural links to Manin triples and double constructions in the theory of Frobenius commutative algebras. They serve as both a categorical and operational enrichment of the perm algebra paradigm, especially at the bialgebra level, admitting special triangular and factorizable subcases closely tied to Yang–Baxter and Rota–Baxter operator theory.

1. Algebraic Definition and Two-Part Splitting

A special Apre-Perm algebra is defined on a vector space $A$ equipped with two bilinear operations $\triangleright$ and $\triangleleft$, subject to the following axioms:

• The sum $$x \circ y := x \triangleright y + x \triangleleft y$$ for all $x,y \in A$ defines a perm algebra structure; i.e., $(A, \circ)$ satisfies

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

• The operation $\triangleleft$ is required to be commutative: $x \triangleleft y = y \triangleleft x$ for all $x, y$.
• The induced dual representation from left and right multiplication by $\triangleright, \triangleleft$, specifically $$(\mathcal{L}_\triangleright^* + \mathcal{R}_\triangleleft^*, -\mathcal{R}_\triangleleft^*, A^*)$$, is equivalent to the adjoint representation of $(A, \circ)$.

A "special Apre-Perm bialgebra" arises when this algebraic structure is combined with a compatible coalgebra structure (usually via a co-opposite splitting on the dual).

This construction is not merely a syntactic refinement but is tied to the averaging process: if $(A, \cdot, P)$ is a commutative associative algebra with an averaging operator $P$,

$x \circ y = P(x) \cdot y$

yields a perm algebra; the two-part splitting then lifts the structure to one where, with additional admissibility conditions on a companion operator $Q$ (such as $P(x)\cdot Q(y) = Q(P(x) \cdot y) = Q(x \cdot Q(y))$), $A$ acquires the structure of a special Apre-Perm algebra (Bai et al., 11 Sep 2025, Zhao et al., 10 Oct 2025).

2. Induction from Averaging Infinitesimal Bialgebras

The passage from averaging commutative algebras to special Apre-Perm bialgebras generalizes at the bialgebra level:

• If $(A, \cdot, \Delta, P, Q)$ is a commutative and cocommutative infinitesimal bialgebra with admissible averaging and companion operators, $P$ and $Q$, then $(A, \triangleright, \triangleleft)$, defined via

$$x \triangleright y = P(x) \cdot y, \qquad x \triangleleft y = Q(x) \cdot y,$$

is a special Apre-Perm algebra.

• The compatible coalgebra structure is constructed via a dual splitting. For instance, letting $\Delta$ be the infinitesimal coproduct,

$$\theta(x) = -\Delta(Px), \qquad \eta(x) = (Q \otimes \mathrm{id})\Delta(x) + \Delta(Px)$$

realizes the corresponding special Apre-Perm bialgebra [(Bai et al., 11 Sep 2025), Proposition 5.9].

Averaging infinitesimal bialgebras admit characterization in terms of double constructions of averaging Frobenius commutative algebras, leading to a Manin triple framework for special Apre-Perm bialgebras [(Bai et al., 11 Sep 2025), Theorem 2.11; (Zhao et al., 10 Oct 2025)].

3. Double Construction and Manin Triple Characterization

A central result is the equivalence between special Apre-Perm bialgebras and certain Manin triples:

• Given a double construction of averaging Frobenius commutative algebras, the direct sum $A \oplus A^*$ can be endowed with a quadratic special Apre-Perm algebra structure with a nondegenerate symmetric left-invariant bilinear form.
• Manin triple: $(A \oplus A^*, \circ_d, B_d)$, with subalgebras $(A, \circ, P)$ and $(A^*, \circ, Q^*)$ forms a Manin triple of special Apre-Perm algebras if and only if $(A, \triangleright, \triangleleft)$ is special Apre-Perm [(Bai et al., 11 Sep 2025), Equations (eq:dc1), (eq:dc2)].

This structural duality makes the special Apre-Perm framework parallel to the classical Lie bialgebra scenario, providing an algebraic setting for the paper of induced, split, and double algebraic objects.

4. Quasi-Triangular, Triangular, and Factorizable Cases

Special subclasses of special Apre-Perm bialgebras arise from solutions to suitable Yang–Baxter equations:

• Quasi-triangular bialgebras are induced by solutions $r$ of the averaging associative Yang–Baxter equation (AAYBE) in $(A, \cdot)$ with invariant symmetric part, i.e.,

$$(\mathrm{id}\otimes L_x - L_x\otimes \mathrm{id})(r + \tau(r)) = 0$$

for all $x \in A$.

• Triangular bialgebras correspond to the case where $r$ is skew-symmetric ($r + \tau(r) = 0$), leading to a simplified operator form and associativity in the induced structure.
• Factorizable bialgebras occur when $r + \tau(r)$ induces an invertible map $A^* \to A$, and any $a\in A$ admits unique decomposition $a = a_+ + a_-$ with $a_+ \in \mathrm{Im}\, r^\sharp$, $a_- \in \mathrm{Im}\, (-\tau(r))^\sharp$ (Zhao et al., 10 Oct 2025).

These special cases are in bijective correspondence with symmetric averaging Rota–Baxter Frobenius commutative algebras and their quadratic versions.

5. Yang–Baxter and $\mathcal{O}$-Operator Characterizations

The operator-theoretic framework aligns the construction of special Apre-Perm bialgebras with the theory of Yang–Baxter and Rota–Baxter operators:

• For a given solution $r$ as above, a corresponding $\mathcal{O}$-operator (of weight $-1$) on $A$ is defined by

$$\langle r^\sharp(a^*), x \rangle = \langle r, a^* \otimes x \rangle,$$

and $r^\sharp$ is an $\mathcal{O}$-operator if and only if $r$ solves the special Apre-Perm Yang–Baxter equation (see [(Zhao et al., 10 Oct 2025), Propositions 4.8, 4.10]).

• Operator identities for $r^\sharp$ are derived from the splitting, e.g.,

$$r^\sharp(a^*) \triangleright r^\sharp(b^*) = r^\sharp( \mathcal{O}_\text{specified}(a^*, b^*) )$$

(and analogously for $\triangleleft$), making these bialgebras amenable to operator-theoretic analysis.

6. Parallel Procedures and Structural Applications

The parallelism between constructions for averaging bialgebras and special Apre-Perm bialgebras is reflected at every level:

• SAPP bialgebras admit operator forms, classification of subclasses (triangular, factorizable), and are characterized via commutative diagrams linking Zinbiel, pre-SAPP, and Rota–Baxter structures [(Zhao et al., 10 Oct 2025), Corollaries 3.12, 3.15].
• Every SAPP bialgebra induces:
  • A pre-Lie algebra by restricting to the $\triangleright$ product.
  • An anti-pre-Lie algebra under the combined product $$x \diamond y = x \triangleright y + 2(x \triangleleft y)$$ [(Bai et al., 11 Sep 2025), Section 4.6].

Practical implications include applications in operad theory, deformation quantization, and the structural paper of algebraically induced bialgebras—especially those that emerge from symmetrization, averaging procedures, or as algebraic models for quantum or topological structures.

7. Summary Table of Key Structures and Correspondences

Structure Type	Key Feature	Reference Section
Special Apre-Perm algebra	Splitting $\circ = \triangleright + \triangleleft$; $\triangleleft$ commutative	(Bai et al., 11 Sep 2025); (Zhao et al., 10 Oct 2025)
Special Apre-Perm bialgebra	Special Apre-Perm algebra + compatible coalgebra	(Bai et al., 11 Sep 2025); (Zhao et al., 10 Oct 2025)
Quasi-triangular SAPP bialgebra	$r$ solves SAPP–YBE; symmetric part invariant	[(Zhao et al., 10 Oct 2025), Prop. 4.8]
Triangular SAPP bialgebra	$r$ skew-symmetric	[(Zhao et al., 10 Oct 2025), Cor. 3.12]
Factorizable SAPP bialgebra	$(r+\tau(r))^\sharp$ invertible	[(Zhao et al., 10 Oct 2025), Cor. 3.15]
Associated operator structures	$\mathcal{O}$-operator, Rota–Baxter correspondence	[(Zhao et al., 10 Oct 2025), Prop. 4.13]
Manin triple equivalence	Double construction via Frobenius averaging algebra	[(Bai et al., 11 Sep 2025), Thm. 5.8]

Special Apre-Perm bialgebras form a natural extension of perm and averaging bialgebra theory, providing a rigorous framework for algebraic structures with refined splitting, operator-theoretic, and cohomological properties. Their paper links Frobenius algebra theory, Yang–Baxter and Rota–Baxter operators, and quantum algebraic structures, with a systematically developed parallel between averaging-induced and split perm-algebraic bialgebra frameworks.