On the irreducible representations of the Jordan triple system of $p \times q$ matrices

TL;DR

This paper characterizes the universal associative envelope of the Jordan triple system of rectangular matrices, showing it is isomorphic to a full matrix algebra and identifying its unique irreducible representation.

Contribution

It establishes the isomorphism of the universal envelope with a matrix algebra and determines the irreducible representations of the Jordan triple system.

Findings

Universal envelope is isomorphic to a full matrix algebra.

There is only one nontrivial irreducible representation.

The center of the universal envelope is identified.

Abstract

Let $\mathcal{J}_{\field}$ be the Jordan triple system of all $p \times q$ ($p\neq q$; $p,q >1)$ rectangular matrices over a field $\field$ of characteristic 0 with the triple product $\{x,y,z\}= x y^t z+ z y^t x $, where $y^t$ is the transpose of $y$. We study the universal associative envelope $\mathcal{U}(\mathcal{J}_{\field})$ of $\mathcal{J}_{\field}$ and show that $\mathcal{U}(\mathcal{J}_{\field}) \cong M_{p+q \times p+q}(\field)$, where $M_{p+q\times p+q} (\field)$ is the ordinary associative algebra of all $(p+q) \times (p+q)$ matrices over $\field$. It follows that there exist only one nontrivial irreducible representation of $\mathcal{J}_{\field}$. The center of $\mathcal{U}(\mathcal{J}_{\field})$ is deduced.