Manin Triple of Special Apre-Perm Algebras

• Manin triple of special apre-Perm algebras is defined by a dual relationship mediated by an invariant symmetric bilinear form and a compatible splitting of perm algebras.
• The framework uses a double construction that integrates averaging operators with commutative splitting to yield robust bialgebra structures.
• This structure extends to operadic and infinite-dimensional contexts, providing practical insights for applications in both classical and quantum algebraic theories.

A Manin triple of special Apre-Perm algebras refers to a highly structured relationship between certain nonassociative algebras (arising as induced objects from perm algebras equipped with averaging operators) and their duals, mediated by an invariant bilinear form and compatible splitting operations. This framework generalizes the classical Manin triple from Lie bialgebra theory and plays a central role in understanding the algebraic and coalgebraic properties of special Apre-Perm algebras, their duals, and extensions to bialgebra theory (Bai et al., 11 Sep 2025).

1. Definition and Construction of Special Apre-Perm Algebras

Apre-Perm algebras emerge from a new splitting of the multiplication in perm algebras. Given a perm algebra $(A, *)$, one introduces a pair of linear operations $(\cdot, \circ)$ so that for $a, b \in A$,

$a * b = a \cdot b + a \circ b$

where the operation $\cdot$ retains the "permutative" property, and $\circ$ is defined via additional representation-theoretic data induced by an averaging operator on a commutative associative algebra.

A special apre-Perm algebra is an apre-Perm algebra satisfying that the second multiplication $\circ$ is commutative, and there exists a nondegenerate symmetric left-invariant bilinear form. Such algebras naturally arise as underlying structures of perm algebras with averaging operators and invariant symmetric Frobenius forms (Bai et al., 11 Sep 2025).

2. Double Construction and Manin Triple Framework

The double construction is a fundamental method for synthesizing Manin triple structures in this context. For a symmetric Frobenius commutative algebra $(F, m, \langle, \rangle)$ with an averaging operator $P$, one considers its double $F \oplus F^*$, with multiplication

$(x + \xi) \star (y + \eta) := x * y + Q(\xi, \eta)$

where $Q(-,-)$ encodes the cross-terms governed by the invariant form and dual module actions. The double is equipped with a symmetric nondegenerate bilinear form

$$B(x + \xi, y + \eta) = \langle x, \eta \rangle + \langle \xi, y \rangle$$

which is left-invariant with respect to both algebra products and intertwines $F$ and $F^*$ as maximal isotropic subalgebras. The triple $(F \oplus F^*, F, F^*)$ with $B$ thus constitutes a Manin triple of special apre-Perm algebras (Bai et al., 11 Sep 2025).

3. Bialgebra Structures Induced from Averaging Operators

Averaging operators on commutative associative algebras (originating in turbulence theory and formalized by Reynolds (Bai et al., 11 Sep 2025)) not only induce perm algebra structures but, when combined with compatible cocommutative infinitesimal bialgebra structures, yield bialgebra variants called special apre-Perm bialgebras. The associated coproduct $\Delta$ is constructed from the averaging operator and the infinitesimal bialgebra comultiplication. The compatibility between $(\cdot, \circ)$ and $\Delta$ is encoded precisely by the Manin triple structure:

• The algebra $A = F$ (with the split product), its dual $A^*$, and their direct sum $A \oplus A^*$, together with $B$, satisfy that $A$, $A^*$ are maximal isotropic with $B$ invariance, and $A \oplus A^*$ has the sum structure of $\cdot$ and $\circ$.
• The bialgebra structure (the "double") is obtained as the direct sum algebra equipped with both products and the comultiplication induced from $A$ and $A^*$ (Bai et al., 11 Sep 2025).

4. Algebraic Splitting and Representation-Theoretic Features

The splitting underlying apre-Perm algebras is fundamentally different from the usual dendriform or pre-Perm splitting. In apre-Perm algebras, the second product is defined by the representation induced from the averaging operator and depends on both the module structure and the invariant Frobenius pairing. The presence of a symmetric Frobenius form ensures the nondegeneracy and maximal isotropy conditions for the subalgebras in any Manin triple.

Special apre-Perm algebras also admit a geometric interpretation as algebraic structures encoding averaging-induced direct sum decompositions; the left-invariance of the bilinear form ensures the compatibility required for a Manin triple.

5. Extension to Infinite-Dimensional and Operadic Contexts

Using graded modules or the Laurent polynomial algebra as the underlying vector space, one constructs infinite-dimensional examples of special apre-Perm algebras and bialgebras. The completed tensor product, as introduced for anti-Leibniz and associated bialgebras (Hou et al., 28 Dec 2024), allows for the extension of Manin triple structures to the infinite-dimensional setting.

Moreover, the operadic perspective is crucial: the Koszul duality between perm and pre-Lie operads (Lin et al., 20 Sep 2024) establishes that the Manin triple framework for special apre-Perm algebras can be extended and examined via operad-theoretic techniques, facilitating connections to pre-Lie, dendriform, and other splitting operads.

6. Historical Development and Applications

The theory of averaging operators dates back to Reynolds' work on turbulence, where averaging was first conceived to separate mean and fluctuating components in fluid flows (Bai et al., 11 Sep 2025). Subsequent algebraic generalizations (by Miller, Rota, and Birkhoff) translated these ideas into commutative and associative algebraic contexts. The connection between averaging operators, symmetric Frobenius forms, and the induced perm and apre-Perm algebra structures is central to recent advances linking fluid dynamics, operad theory, and bialgebra construction. These algebraic objects provide explicit models for the symmetries and dualities inherent in both classical and quantum algebras, and are foundational in the paper of higher gauge theories and algebraic representation theory (Bai et al., 11 Sep 2025).

7. Summary Table of Key Structures

Structure	Defining Features	Role in Manin Triple
Special apre-Perm alg.	$(A, \cdot, \circ)$ with commutative $\circ$ and nondeg. bilinear form	Maximal isotropic subalgebra
Double construction	$A \oplus A^*$ with compatible multiplication and invariant form	Realizes the Manin triple; encodes bialgebra
Averaging operator	Linear operator on commutative associative algebra	Induces splitting and commutative product
Frobenius algebra	Invariant symmetric bilinear form	Provides nondegeneracy condition

A Manin triple of special apre-Perm algebras thus unifies the algebraic and coalgebraic aspects of algebras induced by averaging operators, codifies their double construction, and establishes the foundational bialgebra structure via invariant bilinear forms, maximal isotropy, and compatible algebraic operations (Bai et al., 11 Sep 2025).