Hom-Quadri Dendriform Algebras Overview

• Hom-quadri dendriform algebras are twisted nonassociative structures defined by a linear twisting map and four bilinear operations.
• They are constructed and classified through averaging operators and low-dimensional examples, linking them to Hom-diassociative and Hom-triassociative algebras.
• The framework establishes categorical embeddings and splitting hierarchies in Hom-type algebras, supporting further research in deformation and representation theory.

Hom-quadri-dendriform algebras constitute a family of twisted nonassociative algebraic structures characterized by four bilinear operations and a linear twisting map, forming a splitting framework for twisted diassociative and triassociative algebras. Defined and developed in Hamdouni, Basdouri, Jendoubi, and Abdou's work (Hamdouni et al., 30 Dec 2025), these algebras generalize classical quadri-dendriform algebras to the Hom-type setting, where associativity and related axioms are modified (twisted) via the twisting linear endomorphism. Their study reveals deep categorical relationships, explicit low-dimensional classifications, and systematic constructions through averaging operators.

1. Axiomatic Foundations

A Hom-quadri-dendriform algebra is a quintuple $$(A, \alpha, \nwarrow, \nearrow, \swarrow, \searrow)$$ where $A$ is a vector space over a field $K$, $\alpha:A\to A$ is a linear map, and four bilinear operations $$\nwarrow, \nearrow, \swarrow, \searrow: A\otimes A \to A$$ are defined. The structure is governed by the following eleven twisted identities, which must hold for all $x, y, z \in A$:

$$\begin{align*} \mathrm{(HQ1)}:\quad & (x\nwarrow y)\nwarrow\alpha(z)\;=\;\alpha(x)\nwarrow\bigl(y\nwarrow z + y\nearrow z\bigr) \ \mathrm{(HQ2)}:\quad & (x\nearrow y)\nwarrow\alpha(z)\;=\;\alpha(x)\nearrow(y\nwarrow z) \ \mathrm{(HQ3)}:\quad & \alpha(x)\nearrow(y\nearrow z)\;=\; (x\nwarrow y + x\nearrow y)\nearrow\alpha(z) \ \mathrm{(HQ4)}:\quad & \alpha(x)\nearrow(y\nearrow z)\;=\; (x\swarrow y + x\searrow y)\nearrow\alpha(z) \ \mathrm{(HQ5)}:\quad & (x\nwarrow y)\swarrow\alpha(z)\;=\;\alpha(x)\nwarrow (y\swarrow z + y\searrow z) \ \mathrm{(HQ6)}:\quad & (x\nearrow y)\swarrow\alpha(z)\;=\;\alpha(x)\nearrow(y\swarrow z) \ \mathrm{(HQ7)}:\quad & \alpha(x)\nearrow(y\searrow z)\;=\;(x\nwarrow y + x\nearrow y)\searrow\alpha(z) \ \mathrm{(HQ8)}:\quad & (x\swarrow y)\swarrow\alpha(z)\;=\;\alpha(x)\swarrow(y\nwarrow z + y\nearrow z) = \alpha(x)\swarrow(y\swarrow z + y\searrow z) \ \mathrm{(HQ9)}:\quad & (x\swarrow y)\swarrow\alpha(z)\;=\;\alpha(x)\swarrow(y\nwarrow z + y\searrow z) = \alpha(x)\swarrow(y\swarrow z + y\nearrow z) \ \mathrm{(HQ10)}:\quad & (x\searrow y)\swarrow\alpha(z)\;=\;\alpha(x)\searrow(y\nwarrow z) = \alpha(x)\searrow(y\swarrow z) \ \mathrm{(HQ11)}:\quad & \alpha(x)\searrow(y\nearrow z) = \alpha(x)\searrow(y\searrow z) = (x\swarrow y + x\searrow y)\searrow\alpha(z) \end{align*}$$

The operations are denoted suggestively by compass directions: northwest ($\nwarrow$), northeast ($\nearrow$), southwest ($\swarrow$), and southeast ($\searrow$), corresponding to splittings of the two associative-type products $\vdash$ and $\dashv$ present in Hom-diassociative contexts.

Multiplicativity of the structure requires that $\alpha$ commute with each of the four products: $\alpha(x*y) = \alpha(x)*\alpha(y)$ for each product $* \in \{\nwarrow, \nearrow, \swarrow, \searrow\}$.

2. Splitting Hom-diassociative and Hom-triassociative Structures

Hom-quadri-dendriform algebras are inherently linked to Hom-diassociative and Hom-triassociative algebras via operation splitting. A Hom-diassociative algebra $(D, \dashv, \vdash, \alpha)$ possesses two products and five twisted associativity conditions. Any Hom-quadri-dendriform algebra $$(A, \nwarrow, \nearrow, \swarrow, \searrow, \alpha)$$ yields a Hom-diassociative algebra under

$$x\vdash y := x\nwarrow y + x\nearrow y, \qquad x\dashv y := x\swarrow y + x\searrow y.$$

Conversely, Hom-quadri-dendriform structure can be induced from a Hom-dendriform algebra $(D, \prec, \succ, \alpha)$ equipped with a relative averaging operator $T:V\to D$:

$$u\nwarrow^T v = T(u)\prec v, \quad u\swarrow^T v = u\prec T(v), \quad u\nearrow^T v = T(u)\succ v, \quad u\searrow^T v = u\succ T(v).$$

The graph of $T$ embeds into the hemi–semidirect product Hom-quadri-dendriform algebra $D\oplus V$, and this construction is supported by explicit categorical embeddings (Theorem 3.5 and 3.9 in (Hamdouni et al., 30 Dec 2025)).

3. Categorical and Functorial Relationships

The category of Hom-quadri-dendriform algebras fits into a commutative diagram of categories and functors, capturing the splittings and connecting them to broader classes:

$$\begin{array}{ccc} \{\text{Hom-quadri-dendriform}\} &\longrightarrow& \{\text{Hom-diassociative}\} \ \downarrow &&\downarrow \ \{\text{Hom-dendriform}\} &\longrightarrow& \{\text{Hom-associative}\} \end{array}$$

The forgetful functor $U:\mathbf{HQD}\to\mathbf{HDias}$ sends $(A,\nwarrow,\nearrow,\swarrow,\searrow,\alpha)$ to its Hom-diassociative shadow $(A,\vdash,\dashv,\alpha)$, and has a left adjoint (quadri-enveloping) functor. Similarly, a Hom-dendriform forgetful functor targets Hom-associative algebras via the sum product.

Theorem 3.9 asserts every Hom-quadri-dendriform algebra embeds into an averaging Hom-dendriform algebra using the canonical projection to the dendriform quotient. This formalizes the role of relative averaging operators in the general construction and shows the structural compatibility of quadri-dendriform and dendriform objects in the Hom setting.

Hom-quadri-dendriform and Hom-six-dendriform algebras are situated within a hierarchy, with Hom-triassociative algebras at the top, establishing a layered structure of operation splittings.

4. Explicit Low-dimensional Classification

The full low-dimensional classification of Hom-quadri-dendriform algebras is provided over $\mathbb{C}$ for dimensions two and three. For two dimensions, every algebra is up to isomorphism one of five families ($D_1, \ldots, D_5$), with structure constants regulating the nonzero products and the form of the twisting map $\alpha$:

Model	Nonzero Products (Sample)	Twisting Map $\alpha$
$D_1$	$$e_1\nwarrow e_1=e_2; e_1\nearrow e_1=e_2; e_1\swarrow e_1=e_2; e_1\searrow e_1=\frac12 e_2$$	$\alpha(e_1)=a e_1 + e_2, \alpha(e_2)=a e_2$
$D_2$	$$e_1\nwarrow e_1=e_2; e_1\nearrow e_1=-e_2; e_1\swarrow e_1=e_2; e_1\searrow e_1=e_2$$	$$\alpha=\begin{pmatrix} a & 0 \ 0 & 0 \end{pmatrix}$$
$D_3, D_4(\gamma), D_5(\eta)$	See Theorem 5.1 in (Hamdouni et al., 30 Dec 2025)	Explicit forms by parameters

The three-dimensional classification encompasses thirteen families ($D_1,\ldots,D_{13}$), for example:

$$\begin{aligned} & e_1\nwarrow e_1 = e_2; \quad e_1\nearrow e_1 = e_2; \quad e_1\nwarrow e_3 = e_2; \quad e_1\nearrow e_3 = e_2; \ & e_3\nwarrow e_3 = e_2; \quad e_3\nearrow e_3 = e_2; \quad e_1\swarrow e_1 = e_2; \quad e_1\searrow e_3 = e_2; \ & e_1\swarrow e_3 = e_2; \quad e_3\searrow e_1 = e_2; \quad e_3\swarrow e_3 = e_2; \quad e_3\searrow e_3 = e_2 \end{aligned}$$

$\alpha = aI + N$, with $N^2=0, a\in\mathbb{R}$.

The exhaustive classification for these and additional families is detailed in Section 5 of (Hamdouni et al., 30 Dec 2025), with structure constants for all product operations and twisting maps.

5. Construction via Averaging Operators

A central systematic construction of Hom-quadri-dendriform algebras exploits relative averaging operators. For any Hom-dendriform algebra $(D, \prec, \succ, \alpha)$ and averaging operator $T: D\to D$ (satisfying $T\circ\alpha=\alpha\circ T$ and $T(x)\prec T(y) = T(T(x)\prec y) = T(x\prec T(y))$), define

$$\begin{aligned} & x\nwarrow y = T(x)\prec y \ & x\swarrow y = x\prec T(y) \ & x\nearrow y = T(x)\succ y \ & x\searrow y = x\succ T(y) \end{aligned}$$

These operations fulfill all Hom-quadri-dendriform identities.

For instance, choosing $D = \mathbb{R}^2$ with basis $e_1, e_2$, operations $e_1\prec e_1 = e_2, e_1\succ e_1 = e_2$, $\alpha = \mathrm{Id}$, and $T(e_1) = e_1, T(e_2) = 0$, yields products:

$$e_1\nwarrow e_1 = e_2; \quad e_1\swarrow e_1 = e_2; \quad e_1\nearrow e_1 = e_2; \quad e_1\searrow e_1 = e_2$$

All other products vanish, and the Hom-quadri-dendriform axioms reduce to those of the Hom-dendriform algebra in this example.

This construction shows the role of averaging operators in generating Hom-quadri-dendriform structures from simpler Hom-type algebras and underpins several classification models.

6. Position in the Splitting Hierarchy

Hom-quadri-dendriform algebras and their six-dendriform analogues fit into a layered hierarchy of Hom-algebraic objects, ultimately culminating in Hom-triassociative algebras. The relationships between these algebras are encoded via functorial diagrams and operation splitting, making explicit the passage from general associative-type products down to their finely split dendriform versions. The categorical embeddings and forgetful functors systematically relate Hom-quadri-dendriform, Hom-diassociative, Hom-dendriform, and Hom-associative algebras.

A plausible implication is that further generalizations could pursue higher-order splittings and their Hom-type analogues, echoing the construction and categorical organization demonstrated for quadri-dendriform structures.

7. Principal Developments and Significance

The systematic introduction, explicit axiomatic formulation, and categorical embedding of Hom-quadri-dendriform algebras, as achieved in (Hamdouni et al., 30 Dec 2025), establish a coherent framework for splitting operations in Hom-type algebras. The correspondence with averaging operators, comprehensive low-dimensional classification, and the broader categorical picture clarify their foundational role in the Hom-algebra landscape and provide explicit models for further study in deformation theory, representation theory, and Hom-type nonassociative algebraic systems.