Classification of Low-dimensional Complex Triassociative Algebras

TL;DR

This paper classifies low-dimensional complex triassociative algebras, expanding understanding of their structure by systematically categorizing 1- and 2-dimensional cases.

Contribution

It provides the first comprehensive classification of 1- and 2-dimensional complex triassociative algebras, a generalization of associative dialgebras.

Findings

Complete classification of 1-dimensional algebras

Complete classification of 2-dimensional algebras

Identification of distinct algebraic structures within these dimensions

Abstract

The paper concerns associative trialgebras, also known as triassociative algebras, which were first studied by Loday and Ronco in 2001. These generalize Loday's associative dialgebras (diassociative algebras) and are characterized by 3 operations and 11 identities. The paper details the classification of 1-dimensional and 2-dimensional triassociative algebras over a complex vector space.