Universal associative envelopes of nonassociative triple systems

TL;DR

This paper constructs and analyzes universal associative envelopes for specific nonassociative triple systems, revealing their structure, bases, centers, and representations, and relating infinite cases to down-up algebras.

Contribution

It introduces a method to construct universal associative envelopes for nonassociative triple systems derived from 2-dimensional simple systems, including their structure and representation theory.

Findings

Infinite dimensional envelopes relate to down-up algebras.

Finite dimensional envelopes have determined Wedderburn decompositions.

Irreducible representations are classified for finite cases.

Abstract

We construct universal associative envelopes for the nonassociative triple systems arising from the trilinear operations of Bremner and Peresi applied to the 2-dimensional simple associative triple system. We use noncommutative Gr\"obner bases to determine monomial bases, structure constants, and centers of the universal envelopes. We show that the infinite dimensional envelopes are closely related to the down-up algebras of Benkart and Roby. For the finite dimensional envelopes, we determine the Wedderburn decompositions and classify the irreducible representations.