Representations of simple anti-Jordan triple systems of m × n matrices

Abstract

We show that the universal associative envelope of the simple anti-Jordan triple system of all [Formula: see text] ([Formula: see text] is even, [Formula: see text]) 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 [Formula: see text] matrices is infinite-dimensional.