On Nilpotent Triassociative Algebras

TL;DR

This paper studies nilpotent triassociative algebras, classifying low-dimensional cases and establishing key properties, including a version of Engel's Theorem, revealing how the nilpotency of one operation influences the entire algebra.

Contribution

It provides the first classification and structural analysis of nilpotent triassociative algebras, including a new Engel-type theorem and insights into the behavior of their multiplications.

Findings

One multiplication's nilpotency implies the entire algebra's nilpotency.

The other two multiplications are nilpotent if this particular one is.

A classification of low-dimensional nilpotent triassociative algebras is provided.

Abstract

The class of associative trialgebras, also known as triassociative algebras, is characterized by three multiplications and eleven relations that generalize associativity. In the current paper, we present a study of nilpotent triassociative algebras. After some examples and basic results, we provide a low-dimensional classification, a general monomial form, and an analogue of Engel's Theorem. The main result shows that one of the three multiplication operations behaves differently than the other two. In particular, if this structure alone is nilpotent, then both other multiplications are nilpotent. The converse is not true. Furthermore, the former is nilpotent if and only if the entire algebra is nilpotent.