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

Authors : Hader A ELGENDY

Pages : 213-225

Doi:10.24330/ieja.1226320

View : 21 | Download : 13

Publication Date : 2023-01-09

Article Type : Research Paper

Abstract :Let $\\mathcal{J}_{\\field}$ be the Jordan triple system of all $p \\times q$ insert ignore into journalissuearticles values($p\\neq q$; $p,q >1);$ 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 $\\mathcal{U}insert ignore into journalissuearticles values(\\mathcal{J}_{\\field});$ of $\\mathcal{J}_{\\field}$ and show that $\\mathcal{U}insert ignore into journalissuearticles values(\\mathcal{J}_{\\field}); \\cong M_{p+q \\times p+q}insert ignore into journalissuearticles values(\\field);$, where $M_{p+q\\times p+q} insert ignore into journalissuearticles values(\\field);$ is the ordinary associative algebra of all $insert ignore into journalissuearticles values(p+q); \\times insert ignore into journalissuearticles values(p+q);$ matrices over $\\field$. It follows that there exists only one nontrivial irreducible representation of $\\mathcal{J}_{\\field}$. The center of $\\mathcal{U}insert ignore into journalissuearticles values(\\mathcal{J}_{\\field});$ is deduced.
Keywords : Jordan triple system, rectangular matrix, universal associative envelope, representation theory