The universal envelope of a polarized anti-Jordan triple system of block matrices

Abstract

We investigate the universal associative envelope [Formula: see text] of the simple polarized anti-Jordan triple system [Formula: see text] of [Formula: see text] [Formula: see text] block matrices: [Formula: see text] with the triple product [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic 0. We prove that [Formula: see text], where [Formula: see text] is the ordinary associative algebra of all [Formula: see text] matrices over [Formula: see text]. From this result we classify up to equivalence all finite-dimensional irreducible representations of the anti-Jordan triple system [Formula: see text]. We conclude the paper with an open problem for further research.