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

Abstract

Let $\mathcal{J}_{\field}$ be the Jordan triple system of all $p\times q$ ($p\ne q$; $p,q>1)$ 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.

arXiv commentThis paper has been published in International Electronic Journal of Algebra

References in corpus (1)

• Triple Products and Yang-Baxter Equation (II): Orthogonal and Symplectic Ternary Systems