Classification of tridendriform algebra and related structures

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.