Classification of tridendriform algebra and related structures

TL;DR

This paper classifies tridendriform algebras and their derivations and centroids, revealing the structure and diversity of these algebraic objects across various dimensions using computational methods.

Contribution

It provides the first explicit classification of derivations and centroids of tridendriform algebras up to dimension five, including computational analysis of structure constants.

Findings

Only trivial derivations for 2- and 3-dimensional algebras.

21 non-isomorphic derivations for 4-dimensional algebras.

Number of centroid classes varies with dimension, up to 21 for 4-dimensional algebras.

Abstract

The classification of algebraic structures and their derivations is an important and ongoing research area in mathematics and physics, and various results have been obtained in this field. This article presents the classification of tridendriform algebras that was first studied by Loday and Ronco, including an analysis of structure constant equations using computer algebra software. We further explicitly classify the derivations and centroids of tridendriform algebras, showing that there are only trivial derivations for $2$- and $3$-dimensional algebras but $21$ non-isomorphic derivations for $4$-dimensional tridendriform algebras with dimension range from $1$ to $5$. Additionally, for centroids (centroid and quasi-centroid), there are trivial isomorphism classes for $2$ dimensional tridendriform algebra, $6$ non-isomorphic classes for $3$-dimensional tridendriform algebras and $21$ for $4$-dimensional algebras. The dimensions range for centroid is from $1$ to $5$, whereas it is from $1$ to $10$ for quasi-centroid.