REPRESENTATIONS OF SIMPLE ANTI-JORDAN TRIPLE SYSTEMS OF m× n MATRICES

We show that the universal associative envelope of the simple anti-Jordan triple system of all $mtimes n$ ($m$ is even, $m, ngeq 2$) matrices over an algebraically closed field of characteristic 0 is finite dimensional. The monomial basis and the center of the universal envelope are determined. The explicit decomposition of the universal envelope into matrix algebras is given. The classification of finite dimensional irreducible representations of an anti-Jordan triple system is obtained. The semi-simplicity of the universal envelope is shown. We also show that the universal associative envelope of the simple polarized anti-Jordan triple system of $(n+1)times (n+1)(n> 2 )$ matrices is infinite-dimensional.