The universal Associative envelope of the anti-Jordan triple system of $n \times n$ matrices

TL;DR

This paper proves that the universal associative envelope of the simple anti-Jordan triple system of matrices is finite dimensional, analyzes its structure, and classifies its irreducible representations.

Contribution

It establishes the finite dimensionality of the universal envelope and provides a detailed structural analysis and classification of representations.

Findings

Universal envelope is finite dimensional for all n ≥ 2.

Explicit decomposition into matrix algebras is achieved.

Universal envelope is semisimple and classifies all irreducible representations.

Abstract

We show that the universal associative enveloping algebra of the simple anti-Jordan triple system of all $n \times n$ matrices $(n \ge 2)$ over an algebraically closed field of characteristic 0 is finite dimensional. We investigate the structure of the universal envelope and focus on the monomial basis, the structure constants, and the center. We explicitly determine the decomposition of the universal envelope into matrix algebras. We classify all finite dimensional irreducible representations of the simple anti-Jordan triple system, and show that the universal envelope is semisimple. We also provide an example to show that the universal enveloping algebras of anti-Jordan triple systems are not necessary to be finite-dimensional.