Ternutator Identities

TL;DR

This paper explores the identities satisfied by the ternutator, a ternary operation generalizing the commutator, and discusses their potential in defining new ternary algebraic structures.

Contribution

It presents various forms of ternutator identities and examines their role in establishing ternary algebra frameworks.

Findings

Derived multiple forms of ternutator identities

Established parallels between ternutator identities and Jacobi identity

Discussed potential for defining ternary algebras

Abstract

The ternary commutator or ternutator, defined as the alternating sum of the product of three operators, has recently drawn much attention as an interesting structure generalising the commutator. The ternutator satisfies cubic identities analogous to the quadratic Jacobi identity for the commutator. We present various forms of these identities and discuss the possibility of using them to define ternary algebras.