On Nilpotent Triassociative Algebras

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.