---
title: Hom-Quadri Dendriform Algebras Overview
url: https://www.emergentmind.com/topics/hom-quadri-dendriform-algebras
type: topic
---

# Hom-Quadri Dendriform Algebras Overview

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 [2601.00040], 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 [2601.00040]).


## 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 [2601.00040] | 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 [2601.00040], 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 [2601.00040], 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.

Source: https://www.emergentmind.com/topics/hom-quadri-dendriform-algebras